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Mckinsey approach to problem solving, a guide to the 7-step mckinsey problem solving process.

McKinsey and Company is recognized for its rigorous approach to problem solving. They train their consultants on their seven-step process that anyone can learn.

This resource guides you through that process, largely informed by the McKinsey Staff Paper 66. It also includes a PowerPoint Toolkit with slide templates of each step of the process that you can download and customize for your own use.

In this guide you'll learn:

Overview of the mckinsey approach to problem solving, problem solving process, problem definition.

• Problem Statement

Stakeholder Analysis Worksheet

Structure the problem, hypothesis trees, issue trees, analyses and workplan, synthesize findings, craft recommendations, communicate, distinctiveness practices, harness the power of collaboration, sources and additional reading, request the mckinsey approach to problem solving.

Problem solving — finding the optimal solution to a given business opportunity or challenge — is the very heart of how consultants create client impact, and considered the most important skill for success at McKinsey.

The characteristic “McKinsey method” of problem solving is a structured, inductive approach that can be used to solve any problem. Using this standardized process saves us from reinventing the problem-solving wheel, and allows for greater focus on distinctiveness in the solution. Every new McKinsey associate must learn this method on his or her first day with the firm.

There are four fundamental disciplines of the McKinsey method:

1. Problem definition

A thorough understanding and crisp definition of the problem.

2. The problem-solving process

Structuring the problem, prioritizing the issues, planning analyses, conducting analyses, synthesizing findings, and developing recommendations.

3. Distinctiveness practices

Constructing alternative perspectives; identifying relationships; distilling the essence of an issue, analysis, or recommendation; and staying ahead of others in the problem-solving process.

4. Collaboratio n

Actively seeking out client, customer, and supplier perspectives, as well as internal and external expert insight and knowledge.

Once the problem has been defined, the problem-solving process proceeds with a series of steps:

• Structure the problem
• Prioritize the issues
• Plan analyses
• Conduct analyses
• Synthesize findings
• Develop recommendations

Not all problems require strict adherence to the process. Some steps may be truncated, such as when specific knowledge or analogies from other industries make it possible to construct hypotheses and associated workplans earlier than their formal place in the process. Nonetheless, it remains important to be capable of executing every step in the basic process.

When confronted with a new and complex problem, this process establishes a path to defining and disaggregating the problem in a way that will allow the team to move to a solution. The process also ensures nothing is missed and concentrates efforts on the highest-impact areas. Adhering to the process gives the client clear steps to follow, building confidence, credibility, and long-term capability.

The most important step in your entire project is to first carefully define the problem. The problem definition will serve the guide all of the team’s work, so it is critical to ensure that all key stakeholders agree that it is the right problem to be solving.

The problem definition will serve the guide all of the team’s work, so it is critical to ensure that all key stakeholders agree that it is the right problem to be solving.

There are often dozens of issues that a team could focus on, and it is often not obvious how to define the problem.

In any real-life situation, there are many possible problem statements. Your choice of problem statement will serve to constrain the range of possible solutions.

Constraints can be a good thing (e.g., limit solutions to actions within the available budget.) And constraints can be a bad thing (e.g., eliminating the possibility of creative ideas.) So choose wisely.

The problem statement may ignore many issues to focus on the priority that should be addressed. The problem statement should be phrased as a question, such that the answer will be the solution.

Example scenario – A family on Friday evening :

A mother, a father, and their two teenage children have all arrived home on a Friday at 6 p.m. The family has not prepared dinner for Friday evening. The daughter has lacrosse practice on Saturday and an essay to write for English class due on Monday. The son has theatre rehearsal on both Saturday and Sunday and will need one parent to drive him to the high school both days, though he can get a ride home with a friend.

The family dog, a poodle, must be taken to the groomer on Saturday morning. The mother will need to spend time this weekend working on assignments for her finance class she is taking as part of her Executive MBA. The father plans to go on a 100-mile bike ride, which he can do either Saturday or Sunday. The family has two cars, but one is at the body shop. They are trying to save money to pay for an addition to their house.

Potential problem definitions – A family on Friday evening :

The problem definition should not be vague, without clear measures of success. Rather, it should be a SMART definition:

• Action-oriented

Given one set of facts, it is possible to come up with many possible problem statements. The choice of problem statement constrains the range of possible solutions.

Before starting to solve the problem, the family first needs to agree on what problem they want to solve.

• What should the family do for dinner on Friday night?
• How can the family schedule their activities this weekend to accomplish everything planned given that they only have one vehicle available?
• How can the family increase income or reduce expenses to allow them to save $75K over the next 12 months to pay for the planned addition to their house?

Problem Statement Worksheet

This is a helpful tool to use to clearly define the problem. There are often dozens of issues that a team could focus on, and it is often not obvious how to define the problem. In any real-life situation, there are many possible problem statements. Your choice of problem statement will serve to constrain the range of possible solutions.

• Use a question . The problem statement should be phrased as a question, such that the answer will be the solution. Make the question SMART: specific, measurable, action-oriented, relevant, and time-bound. Example: “How can XYZ Bank close the $100 million profitability gap in two years?”
• Context . What are the internal and external situations and complications facing the client, such as industry trends, relative position within the industry, capability gaps, financial flexibility, and so on?
• Success criteria . Understand how the client and the team define success and failure. In addition to any quantitative measures identified in the basic question, identify other important quantitative or qualitative measures of success, including timing of impact, visibility of improvement, client capability building required, necessary mindset shifts, and so on.
• Scope and constraints . Scope most commonly covers the markets or segments of interest, whereas constraints govern restrictions on the nature of solutions within those markets or segments.
• Stakeholders . Explore who really makes the decisions — who decides, who can help, and who can block.
• Key sources of insight . What best-practice expertise, knowledge, and engagement approaches already exist? What knowledge from the client, suppliers, and customers needs to be accessed? Be as specific as possible: who, what, when, how, and why.

In completing the Problem Statement Worksheet, you are prompted to define the key stakeholders.

As you become involved in the problem-solving process, you should expand the question of key stakeholders to include what the team wants from them and what they want from the team, their values and motivations (helpful and unhelpful), and the communications mechanisms that will be most effective for each of them.

Using the Stakeholder Analysis Worksheet allows you to comprehensively identify:

• Stakeholders
• What you need from them
• Where they are
• What they need from you

The two most helpful techniques for rigorously structuring any problem are hypothesis trees and issue trees. Each of these techniques disaggregates the primary question into a cascade of issues or hypotheses that, when addressed, will together answer the primary question.

A hypothesis tree might break down the same question into two or more hypotheses. 

The aim at this stage is to structure the problem into discrete, mutually exclusive pieces that are small enough to yield to analysis and that, taken together, are collectively exhaustive.

Articulating the problem as hypotheses, rather than issues, is the preferred approach because it leads to a more focused analysis of the problem. Questions to ask include:

• Is it testable – can you prove or disprove it?
• It is open to debate? If it cannot be wrong, it is simply a statement of fact and unlikely to produce keen insight.
• If you reversed your hypothesis – literally, hypothesized that the exact opposite were true – would you care about the difference it would make to your overall logic?
• If you shared your hypothesis with the CEO, would it sound naive or obvious?
• Does it point directly to an action or actions that the client might take?

Quickly developing a powerful hypothesis tree enables us to develop solutions more rapidly that will have real impact. This can sometimes seem premature to clients, who might find the “solution” reached too quickly and want to see the analysis behind it.

Take care to explain the approach (most important, that a hypothesis is not an answer) and its benefits (that a good hypothesis is the basis of a proven means of successful problem solving and avoids “boiling the ocean”).

Example: Alpha Manufacturing, Inc.

Problem Statement: How can Alpha increase EBITDA by $13M (to $50M) by 2025?

The hypotheses might be:

• Alpha can add $125M revenues by expanding to new customers, adding $8M of EBITDA
• Alpha can reduce costs to improve EBITDA by $5M

These hypotheses will be further disaggregated into subsidiary hypotheses at the next level of the tree.

Often, the team has insufficient knowledge to build a complete hypothesis tree at the start of an engagement. In these cases, it is best to begin by structuring the problem using an issue tree.

An issue tree is best set out as a series of open questions in sentence form. For example, “How can the client minimize its tax burden?” is more useful than “Tax.” Open questions – those that begin with what, how, or why– produce deeper insights than closed ones. In some cases, an issue tree can be sharpened by toggling between issue and hypothesis – working forward from an issue to identify the hypothesis, and back from the hypothesis to sharpen the relevant open question.

Once the problem has been structured, the next step is to prioritize the issues or hypotheses on which the team will focus its work. When prioritizing, it is common to use a two-by-two matrix – e.g., a matrix featuring “impact” and “ease of impact” as the two axes.

Applying some of these prioritization criteria will knock out portions of the issue tree altogether. Consider testing the issues against them all, albeit quickly, to help drive the prioritization process.

Once the criteria are defined, prioritizing should be straightforward: Simply map the issues to the framework and focus on those that score highest against the criteria.

As the team conducts analysis and learns more about the problem and the potential solution, make sure to revisit the prioritization matrix so as to remain focused on the highest-priority issues.

The issues might be:

• How can Alpha increase revenue?
• How can Alpha reduce cost?

Each of these issues is then further broken down into deeper insights to solutions.

If the prioritization has been carried out effectively, the team will have clarified the key issues or hypotheses that must be subjected to analysis. The aim of these analyses is to prove the hypotheses true or false, or to develop useful perspectives on each key issue. Now the task is to design an effective and efficient workplan for conducting the analyses.

Transforming the prioritized problem structure into a workplan involves two main tasks:

• Define the blocks of work that need to be undertaken. Articulate as clearly as possible the desired end products and the analysis necessary to produce them, and estimate the resources and time required.
• Sequence the work blocks in a way that matches the available resources to the need to deliver against key engagement milestones (e.g., important meetings, progress reviews), as well as to the overall pacing of the engagement (i.e., weekly or twice-weekly meetings, and so on).

A good workplan will detail the following for each issue or hypothesis: analyses, end products, sources, and timing and responsibility. Developing the workplan takes time; doing it well requires working through the definition of each element of the workplan in a rigorous and methodical fashion.

It’s useful to match the workplan to three horizons:

• What is expected at the end of the engagement
• What is expected at key progress reviews
• What is due at daily and/or weekly team meetings

The detail in the workplan will typically be greater for the near term (the next week) than for the long term (the study horizon), especially early in a new engagement when considerable ambiguity about the end state remains.

Here are three different templates for a workplan:

This is the most difficult element of the problem-solving process. After a period of being immersed in the details, it is crucial to step back and distinguish the important from the merely interesting. Distinctive problem solvers seek the essence of the story that will underpin a crisp recommendation for action.

Although synthesis appears, formally speaking, as the penultimate step in the process, it should happen throughout. Ideally, after you have made almost any analytical progress, you should attempt to articulate the “Day 1” or “Week 1” answer. Continue to synthesize as you go along. This will remind the team of the question you are trying to answer, assist prioritization, highlight the logical links of the emerging solution, and ensure that you have a story ready to articulate at all times during the study.

McKinsey’s primary tool for synthesizing is the pyramid principle. Essentially, this principle asserts that every synthesis should explain a single concept, per the “governing thought.” The supporting ideas in the synthesis form a thought hierarchy proceeding in a logical structure from the most detailed facts to the governing thought, ruthlessly excluding the interesting but irrelevant.

While this hierarchy can be laid out as a tree (like with issue and hypothesis trees), the best problem solvers capture it by creating dot-dash storylines — the Pyramid Structure for Grouping Arguments.

Pyramid Structure for Grouping Arguments

• Focus on action. Articulate the thoughts at each level of the pyramid as declarative sentences, not as topics. For example, “expansion” is a topic; “We need to expand into the European market” is a declarative sentence.
• Use storylines. PowerPoint is poor at highlighting logical connections, therefore is not a good tool for synthesis. A storyline will clarify elements that may be ambiguous in the PowerPoint presentation.
• Keep the emerging storyline visible. Many teams find that posting the storyline or story- board on the team-room wall helps keep the thinking focused. It also helps in bringing the client along.
• Use the situation-complication-resolution structure. The situation is the reason there is action to be taken. The com- plication is why the situation needs thinking through – typically an industry or client challenge. The resolution is the answer.
• Down the pyramid: does each governing thought pose a single question that is answered completely by the group of boxes below it?
• Across: is each level within the pyramid MECE?
• Up: does each group of boxes, taken together, provide one answer – one “so what?” – that is essentially the governing thought above it?
• Test the solution. What would it mean if your hypotheses all came true?

It is at this point that we address the client’s questions: “What do I do, and how do I do it?” This means not offering actionable recommendations, along with a plan and client commitment for implementation.

The essence of this step is to translate the overall solution into the actions required to deliver sustained impact. A pragmatic action plan should include:

• Relevant initiatives, along with a clear sequence, timing, and mapping of activities required
• Clear owners for each initiative
• Key success factors and the challenges involved in delivering on the initiatives

Crucial questions to ask as you build recommendations for organizational change are:

• Does each person who needs to change (from the CEO to the front line) understand what he or she needs to change and why, and is he or she committed to it?
• Are key leaders and role models throughout the organization personally committed to behaving differently?
• Has the client set in place the necessary formal mechanisms to reinforce the desired change?
• Does the client have the skills and confidence to behave in the desired new way?

Once the recommendations have been crafted in the problem-solving process, it’s vital to effectively communicate those findings and recommendations.

An executive summary is a great slide to use for this. See more on executive summary slides, including 30 templates, at our Ultimate Guide to Executive Summary Slides .

Great problem solvers identify unique disruptions and discontinuities, novel insights, and step-out opportunities that lead to truly distinctive impact. This is done by applying a number of practices throughout the problem-solving process to help develop these insights.

Expand: Construct multiple perspectives

Identifying alternative ways of looking at the problem expands the range of possibilities, opens you up to innovative ideas, and allows you to formulate more powerful hypotheses. Questions that help here include:

• What changes if I think from the perspective of a customer, or a supplier, or a frontline employee, or a competitor?
• How have other industries viewed and addressed this same problem?
• What would it mean if the client sought to run the company like a low-cost airline or a cosmetics manufacturer?

Link: Identify relationships

Strong problem solvers discern connections and recognize patterns in two different ways:

• They seek out the ways in which different problem elements – issues, hypotheses, analyses, work elements, findings, answers, and recommendations – relate to one another.
• They use these relationships throughout the basic problem-solving process to identify efficient problem-solving approaches, novel solutions, and more powerful syntheses.

Distill: Find the essence

Cutting through complexity to identify the heart of the problem and its solution is a critical skill.

• Identify the critical problem elements. Are there some issues, approaches, or options that can be eliminated completely because they won’t make a significant difference to the solution?
• Consider how complex the different elements are and how long it will take to complete them. Wherever possible, quickly advance simpler parts of the problem that can inform more complex or time-consuming elements.

Lead: Stay ahead/step back

Without getting ahead of the client, you cannot be distinctive. Paradoxically, to get ahead – and stay ahead – it is often necessary to step back from the problem to validate or revalidate the approach and the solution.

• Spend time thinking one or more steps ahead of the client and team.
• Constantly check and challenge the rigor of the underlying data and analysis.
• Stress-test the whole emerging recommendation
• Challenge the solution against a set of hurdles. Does it satisfy the criteria for success as set out on the Problem Statement Worksheet?

No matter how skilled, knowledgeable, or experienced you are, you will never create the most distinctive solution on your own. The best problem solvers know how to leverage the power of their team, clients, the Firm, and outside parties. Seeking the right expertise at the right time, and leveraging it in the right way, are ultimately how we bring distinctiveness to our work, how we maximize efficiency, and how we learn.

When solving a problem, it is important to ask, “Have I accessed all the sources of insight that are available?” Here are the sources you should consider:

• Your core team
• The client’s suppliers and customers
• Internal experts and knowledge
• External sources of knowledge
• Communications specialists

The key here is to think open, not closed. Opening up to varied sources of data and perspectives furthers our mission to develop truly innovative and distinctive solutions for our clients.

• McKinsey Staff Paper 66 — not published by McKinsey but possibly found through an internet search
• The McKinsey Way , 1999, by Ethan M. Rasiel

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The Pyramid Model

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A pyramid is a 3-dimensional geometric solid . It consists of a base that is a polygon and a point not on the plane of the polygon, called the vertex . The edges of the pyramid are the sides of the polygonal base together with line segments which join the vertex of the pyramid to each vertex of the polygon.

Some well-known pyramids include the tetrahedron , which has a triangle for its base. (A regular tetrahedron has all edges of equal length, and is one of the Platonic solids ). Another is the regular square pyramid. Two of these with their bases joined form an octahedron , which is another Platonic solid.

Introductory

Intermediate

• Corners are sliced off a unit cube so that the six faces each become regular octagons . What is the total volume of the removed tetrahedra? ( 2007 AMC 12A Problems/Problem 20 )
• In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. Find the ratio of the volume of the smaller tetrahedron to that of the larger. ( 2003 AIME II, #4 )
• A square pyramid with base ABCD and vertex E has eight edges of length 4. A plane passes through the midpoints of AE, BC, and CD. Find the area of the plane's intersection with the pyramid. ( 2007 AIME I, #13 )
• Prove that pyramid volume is 1/3 bh using parallelipiped and dimensional analysis.

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Pyramid problems.

Surface area and volume of pyramid problems along with with detailed solutions are presented.

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• Problems on Pyramid

Solved word problems on pyramid are shown below using step-by-step explanation with the help of the exact diagram in finding surface area and volume of a pyramid.

Worked-out problems on pyramid: 1. The base of a right pyramid is a square of side 24 cm. and its height is 16 cm.

(i) the area of its slant surface

(ii) area of its whole surface and

(iii) its volume. 

Let, the square WXYZ be the base of the right pyramid and its diagonals WY and XZ intersect at O. If  OP  be perpendicular to the plane of the square at O, then  OP  is the height of the pyramid.

Draw  OE  ┴  WX Then, E is the mid - point of  WX . 

By question,  OP  = 16 cm. and  WX  = 24 cm.  Therefore,  OE  =  EX  = 1/2 ∙  WX  = 12 cm Clearly,  PE  is the slant height of the pyramid.  Since  OP  ┴  OE , hence from ∆ POE we get,  PE² = OP² + OE² 

or,PE² = 16² + 12² 

or, PE² = 256 + 144 

or, PE² = 400

PE  = √400

Therefore,  PE  = 20.  Therefore, (i) the required area of slant surface of the right pyramid

= 1/2 × perimeter of the base × slant height. 

= 1/2 × 4 × 24× 20 square cm. 

= 960 square cm. 

(ii) The area of the whole surface of the right pyramid = area of slant surface + area of the base

= (960 + 24 × 24) square cm

= 1536 square cm.

(iii) the volume of the right pyramid

= 1/3 × area of the base × height

= 1/3 × 24 × 24 × 16 cubic cm 

= 3072 cubic cm.

2. The base of a right pyramid 8 m high, is an equilateral triangle of side 12√3 m. Find its volume and the slant surface. Solution:

Let equilateral ∆ WXY be the base and P, the vertex of the right pyramid.

In the plane of the ∆ WXY draw YZ perpendicular to WX and let OZ = 1/3 YZ . Then, O is the centroid of ∆ WXY. Let OP be perpendicular to the plane of ∆ WXY at O; then OP is the height of the pyramid. By question, WX = XY = YW = 8√3 m and OP = 8 m. Since ∆ WXY is equilateral and YZ ┴ WX Hence, Z bisects WX .

Therefore, XZ = 1/2 ∙ WX = 1/2 ∙ 12√3 = 6√3 m. Now, from right - angled ∆ XYZ we get,

YZ² = XY² - XZ²

or, YZ² = (12√3) ² - (6√3)²

or, YZ² = 6² (12 - 3)

or, YZ² = 6² ∙ 9

or, YZ² = 324

Therefore, YZ = 18

Therefore, OZ = 1/3 ∙ 18 = 6. Join PZ . Then, PZ is the slant height of the pyramid. Since OP is perpendicular to the plane of ∆ WXY at O, hence OP ┴ OZ . Therefore, from the right angled ∆ POZ we get,

PZ² = OZ² + OP²

or, PZ ² = 6² + 8²

or, PZ² = 36 + 64

or, PZ² = 100

Therefore, PZ = 10 Therefore, the required slant surface of the right pyramid

= 1/2 × perimetre of the base × slant height

= 1/2 × 3 × 12√3 × PZ

= 1/2 × 36√3 × 10

= 180√3 square meter.

and its volume = 1/3 × area of the base × height

= 1/3 × (√3)/4 (12√3)² × 8

[Since, area of equilateral triangle

= (√3)/4 × (length of a side)² and height = OP = 8]

= 288√3 cubic meter.

●  Mensuration

• Formulas for 3D Shapes
• Volume and Surface Area of the Prism
• Worksheet on Volume and Surface Area of Prism
• Volume and Whole Surface Area of Right Pyramid
• Volume and Whole Surface Area of Tetrahedron
• Volume of a Pyramid
• Volume and Surface Area of a Pyramid
• Worksheet on Volume and Surface Area of a Pyramid
• Worksheet on Volume of a Pyramid

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Here we will learn about pyramids, including what a pyramid is, how to calculate the volume of a pyramid and how to calculate the surface area of a pyramid.

There are also pyramid worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is a pyramid?

A pyramid is a three dimensional shape made up of flat faces: a polygon base and triangular lateral faces which meet at a point, called the apex . A triangular pyramid can also be called a tetrahedron, we can have different types of pyramids such as square based pyramids (like the Egyptian Pyramids), pentagonal pyramids, hexagonal pyramids etc. The base of the pyramids does not need to be a regular polygon. The base can be an irregular polygon and if the apex is not over the centre of the polygon, we call the pyramid and oblique pyramid.

Is a cone a pyramid?

A circle is not a polygon as a polygon has a finite number of edges, connected by vertices. As a pyramid has a polygonal base, the base cannot be a circle as a circle is not a polygon. So, a cone is not a pyramid.

However, the volume of a cone is still equal to V=\frac{1}{3}Bh where B=\pi{r^2}

To calculate the volume or surface area of a pyramid, we need to know certain lengths within the pyramid. Let’s look at each case separately.

Step-by-step guide: Square based pyramid

See also: Triangular based pyramid

Volume of a pyramid

The formula for the volume of a pyrimad is same no matter if it is a regular pyramid or an irregular pyramid.

To calculate the volume of a pyramid , we use the formula:

• V represents the volume of the pyramid,
• B represents the area of the base of the pyramid,
• h represents the perpendicular height of the pyramid.

This formula can be applied to any pyramid where the base is a polygon .

Calculate the volume of the following square based pyramid:

Step-by-step guide: Volume of a pyramid

Surface area of a pyramid

As the area of each triangular face may be different, depending on each side length of the base, the formula for the surface area is a little more tricky to generalise.

However, we can still apply our knowledge of surface area to say:

The surface area of a pyramid is the total sum of the area of each face of the pyramid.

For a square based pyramid, we calculate the area of the base (a square) and the area of four identical triangles:

Note: you may need to calculate the missing length of a triangular side or slant height of the pyramid using Pythagoras’ theorem .

Calculate the surface area of the following square based pyramid:

Step-by-step guide: Surface area of a pyramid

How to calculate the volume of a pyramid

In order to calculate the volume of a pyramid:

Calculate the area of the base.

Substitute values into the formula and solve.

Write the answer, including the units.

Volume and surface area of pyramids worksheet

Get your free pyramid worksheet of 20+ volume and surface area of pyramids questions and answers. Includes reasoning and applied questions.

Pyramid examples

Example 1: volume of a square-based pyramid.

Calculate the volume of the square based pyramid below. Write your answer to 2 decimal places.

The area of the base is:

2 Substitute values into the formula and solve.

The formula for the volume of a pyramid is: V=\frac{1}{3}Bh with B=27.04cm^2 and h=6.8cm . Substituting these values for the base and the vertical height, we have:

3 Write the answer, including the units.

Step-by-step guide: Volume of square based pyramid

Example 2: volume of a rectangular-based pyramid

The pyramid ABCDE has a rectangular base. The apex is vertically above the centre of the base. Calculate the volume of the pyramid, correct to 1 decimal place.

\begin{aligned} &6.8\times 7.2=48.96 \\\\ &B=48.96cm^2 \end{aligned}

The formula for the volume of a pyramid is V=\frac{1}{3}Bh , where B=48.96cm^2 and h=8.1cm . Substituting these values into the formula, we have

\begin{aligned} V&=\frac{1}{3}\times{48.96}\times{8.1}\\\\ V&=132.192 \end{aligned}

Example 3: calculating the height, given the volume

Find the height of a pyramid with volume 48 \ cm^3 and base area 15 \ cm^2 .

We already know that the area of the base is 15 \ cm^2 , so we can move on to step 2 .

We need to use the formula for the volume of a pyramid and solve it to find the value of the height, h .

\begin{aligned} V&=\frac{1}{3}\times{B}\times{h}\\\\ 48&=\frac{1}{3}\times{15}\times{h}\\\\ 48&=5h\\\\ h&=9.6 \end{aligned}

How to calculate the surface area of a pyramid

In order to calculate the surface area of a pyramid:

Calculate the area of all of the triangular faces.

Calculate the total surface area.

Surface area of a pyramid examples

Example 4: finding the surface area from a 3 d diagram.

The polyhedron ABCDE is a square based pyramid. The point M lies on the edge AD where AM=MD . The point E is perpendicular to the centre of the base ABCD .

Calculate the surface area of the pyramid.

5\times 5=5^2=25

The triangular face AED has a height of 7cm, and a width of 5cm. The area of a triangle is A=\frac{1}{2}\times{b}\times{h} , so the area of AED is:

\begin{aligned} A&=\frac{1}{2}\times{b}\times{h} \\\\ &=\frac{1}{2}\times{5}\times{7} \\\\ &=17.5\text{ cm}^{2} \end{aligned}

Each triangular face for a square based pyramid is congruent when the apex is directly above the centre of the base. This means that we can calculate the area of one of the triangular faces, then multiply the solution by 4 to calculate the area of all four triangles.

17.5\times{4}=70\text{ cm}^{2}

The surface area ( SA ) of the pyramid is the total sum of the area of each face, giving us:

SA=25+70=95

Example 5: finding the surface area from a 3 D diagram

ABCD is a triangular based pyramid.

Triangle ABC is equilateral. All of the other faces are congruent, isosceles triangles.

The point M is the midpoint of the line AC such that:

• MB=3\sqrt{3}\text{ cm}
• MC=3\text{ cm}
• MD=8\text{ cm}.

Calculate the surface area of the pyramid. Write your answer in the form a+b\sqrt{c}\text{ cm}^{2} where a, \ b, and c are integers to be found.

The area of the base (which is a triangle) is:

\begin{aligned} A&=\frac{1}{2}\times{b}\times{h}\\\\ &=\frac{1}{2}\times{6}\times{3\sqrt{3}}\\\\ &=9\sqrt{3} \end{aligned}

The remaining three triangular faces are all the same as stated in the question (they are congruent). This means that we can calculate the area of one of the faces, and then multiply it by 3.

The triangle ACD has a height of 8cm , and a width of 6cm (double 3cm ).

The area of triangle ACD is therefore equal to

\begin{aligned} A&=\frac{1}{2}\times{b}\times{h}\\\\ &=\frac{1}{2}\times{8}\times{6}\\\\ &=24\text{ cm}^{2} \end{aligned}

The total surface area is the sum of each of the four triangles:

9\sqrt{3}+3\times 24

As the question asks us to write the solution in the form a+b\sqrt{c}\text{ cm}^{2} , we have:

SA=72+9\sqrt{3}\text{ cm}^{2}

Example 6: finding the surface area from a net

Calculate the surface area of the pyramid. Give your answer correct to 2 decimal places.

The area of the square base is:

7\times 7=7^2=49

We need to add the area of the base to the area of the triangular faces.

There are 4 triangular faces.

First we need to find the height, h , of one of the triangles using Pythagoras theorem.

Now that we know the height of a triangular face, the area of the same triangular face is:

\begin{aligned} A&=\frac{1}{2}\times{7}\times{8.291561976}\\\\ &=29.02046692 \end{aligned}

The area of all four triangular faces is therefore:

29.02046692\times{4}=116.0818677.

Notice that we have not rounded any solution so far (we leave this until the very last line of calculation).

The total surface area of the net is:

49+116.0818677=165.0818677

Common misconceptions

• Use the perpendicular height

The height of the shape needs to be the perpendicular height.  To find the volume of the pyramid we use its perpendicular height.  To find the area of a triangular face we use the perpendicular height of the triangular face. It is also common to confuse the volume of a pyramid with the volume of a prism. It is important to know the difference between 3D shapes.

• Surface area has square units

The surface area will have square units such as square centimetres ( cm^2 ) or square metres ( m^2 ).

• Volume has cube units

The volume will have cube units such as cubic centimetres ( cm^3 ) or cubic metres ( m^3 ).

• Be accurate

When there are two or more steps in your workings, do not round your workings. Only round at the end of your solution so that your answer is accurate.

• Take care with rounding

At the end of the question, make sure you round your answer to the correct number of decimal places or significant figures.

Practice pyramid questions

1. Calculate the volume of the square based pyramid. Write your answer to 2 decimal places.

2. Calculate the volume of the pyramid below.

3. A pyramid has a volume of 160 \ cm^3 and the area of its base is 40 \ cm^2 . Work out the height, h , of the pyramid:

4. PQRST is a square based pyramid. The point M is the midpoint of the line RS such that RM:MS=1:1 . Calculate the surface area of the pyramid.

5. ABCDE is a square based pyramid. E is vertically above the centre of the base of the pyramid. M is the midpoint of the line AD. \ AM=4.15m and EM=10.7m.

Calculate the surface area of the pyramid. Give your answer correct to 1 decimal place.

6. Calculate the surface area of the pyramid.

First we need to find the height of the triangular face.

Pyramid GCSE questions

1. A square-based pyramid has a base with side 15 \ cm.

The perpendicular height of the pyramid is 20 \ cm.

Calculate the volume of the pyramid.

2. The diagram shows a square-based pyramid.

3. A solid is made from a pyramid on top of a cuboid.  AD=12m,CD=10m, CH=9m . M and N are midpoints of the lines EH and EF respectively with IM=8m, and IN=\sqrt{53}m.

Calculate the surface area of the shape. Write your answer to 2 decimal places.

Learning checklist

You have now learned how to:

• Calculate surface areas and volumes of pyramids

The next lessons are

• Alternate angles
• Volume of a cuboid
• Converting units of area and volume
• Pythagoras theorem in 3 dimensions

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

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The Minto Pyramid Principle: Logic in Writing, Thinking, & Problem Solving Expanded Edition

• ISBN-10 0960191038
• ISBN-13 978-0960191031
• Edition Expanded
• Publisher Minto Intl
• Publication date January 1, 1996
• Language English
• Print length 254 pages
• See all details

Product details

• Publisher ‏ : ‎ Minto Intl; Expanded edition (January 1, 1996)
• Language ‏ : ‎ English
• Hardcover ‏ : ‎ 254 pages
• ISBN-10 ‏ : ‎ 0960191038
• ISBN-13 ‏ : ‎ 978-0960191031
• Item Weight ‏ : ‎ 1.98 pounds
• #262 in Business Writing Skills (Books)
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About the author

Barbara minto.

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How SCQA and the Pyramid Principle Fit Into The Strategy Consulting Process

If you are solving a problem using the consulting process, you will leverage the SCQA framework and the Pyramid Principle . In the way I teach students and clients about the consulting process, it involves shifting back and forth between the top-down modes and the bottom-up modes of analysis.

SCQA will typically play a bigger role in the top-down mode, while Part 1 of the Pyramid Principle will help you in the bottom-up mode, and then Part 2 will help you start thinking about how you may communicate your findings.

We can break the consulting process into five high-level steps:

#1 First, we define the problem!

We first start by defining the problem and making sure we are solving the right problem and not just a symptom of a deeper issue. To truly define a problem you want to gather as much data as possible. If you can do this in a team setting, it will help you challenge each other’s perspectives and better understand what’s going on. (To go deeper with defining a problem, you can also turn it into an issue tree .)

We use this to start thinking about a high-level approach to our problem, but not jump to any conclusions.

#2 Do some digging (Initial Research)

Once we define the problem, we will shift back into the bottom-up mode and poke around in the information we have. If it’s a very data heavy type of project, this might involve a few hours of familiarizing yourself with a data set.

While doing this, you want to ask yourself, “What themes are standing out?” and “How is this impacting my initial conception of the problem?”

We’ll use the grouping and insight process of the pyramid principle to make sense of the information we have gathered.

#3 Formulate Your Questions & Hypotheses

From our initial digging into the problem, we likely have landed on several questions worth exploring, some of which may be different than the initial problem that we defined. This will be the first time you start structuring your overall project into a group of questions or hypotheses. Remember the rule of three here to keep it simple for yourself. Grouping the questions will help you split up the work if you are in a team or at minimum to help you focus your attention in different areas.

#4 The Hard (and frustrating) Part – Fine-Tuning

Finding the “answers” to your questions is the next part and while it may seem straightforward it can be a frustrating process. As you start to prove or disprove your hypotheses, you may find a few things happening:

• Your problem statement needs to be tweaked
• Your hypotheses may not be the right questions
• Your hunches about the direction of the findings are off and you need to rethink your approach to the research
• You are realizing that one factor in your research is much more important than you realized and may need deeper research

All of these are to be expected. You’ll want to get feedback from teammates and keep “pressure-testing” your argument to what an outside audience might say. The key is to refine your thinking as soon as you get new information rather than clinging to a certain way of framing the information.

Towards the end of this phase, you’ll start to have a good understanding of the key themes of your research. This is the best time to start drafting an outline of your takeaways using the top-down pyramid principle approach.

You can do this by moving around full-developed or half-built slides as well as drafting an outline for yourself and seeing if you can iterate and find better ways of ordering your information.

Pro Tip: An executive summary is a great internal tool to write out your overall message and see if it makes sense to you. This helps you start to see flaws in the overall message.

#5 Telling The Story

Once you have done the final research to help fill the holes in your story, you’ll want to write a final outline of your “story” – the main takeaway followed by your three main insights or arguments.

You’ll want to spend a lot of time thinking about your audience. What will resonate with them? What method of storytelling – direct or indirect – is best? What type of document to create?

To go deeper you can explore part two of the pyramid principle or twenty persuasion tricks for presentations .

Do you have a toolkit for business problem solving? I created Think Like a Strategy Consultant as an online course to make the tools of strategy consultants accessible to driven professionals, executives, and consultants. This course teaches you how to synthesize information into compelling insights, structure your information in ways that help you solve problems, and develop presentations that resonate at the C-Level. Click here to learn more or if you are interested in getting started now, enroll in the self-paced version ($497) or hands-on coaching version ($997). Both versions include lifetime access and all future updates.

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Pyramid Model keeps behavior problems to a minimum — and kids in school

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Sep 27, 2018, 9:03 PM

by Jane Sevier

When 4-year-old Antonio felt frustrated at home or at school, he used to bite. And hit. And kick. In many schools, that behavior would get him sent home.  

But Antonio is lucky. He attends Cambridge Early Learning Center, a Nashville pre-K center grounded in the principles of the Pyramid Model for Supporting Social Emotional Competence in Infants and Young Children, supported by a $1 million federal grant for preschool development. At Cambridge, Antonio is learning how to communicate his emotions appropriately, engage in social problem-solving and look for solutions other than aggression to get what he wants.

Mary Louise (“M.L.”) Hemmeter , MEd’87, PhD’91, professor of special education at Peabody, was instrumental in developing the Pyramid Model and is part of a team that received $14 million in federal grants last year alone to promote its use. She currently helms a series of projects funded by the U.S. Department of Education and Department of Health and Human Services, and has spent two decades studying effective instruction, social-emotional development and challenging behavior, as well as coaching teachers. “We have to create environments that make children want to be there, where they’re happy and engaged,” Hemmeter says. “If we develop a plan around a child with behavior challenges, the strategies that we use are embedded and used throughout the day. That child is getting individualized support, but it won’t look like discipline in any way.”

Suspensions are commonly used in K–12 settings when behavior problems arise. But they are also common at the preschool level, and unfortunately, developmental and racial disparities exist in these practices. What’s more, expulsions early in a child’s life predict later expulsions and suspensions, which make the child 10 times more likely to drop out of high school, fail academically, feel negatively about school and face incarceration than those who were never suspended or expelled. This discipline gap is a driving influence for Hemmeter.  

“After we have worked with teachers and supported them around the Pyramid Model, they have the strategies and tools that they need,” Hemmeter says. “In the Pyramid Model training, we don’t encourage strategies like ‘time out’ or sending a child to the office. Those strategies don’t teach the child the skills they need; rather they remove the child from the context where they are likely to learn how to get along with others and communicate their emotions.”

Teachers and staff at Cambridge are trained to use the Pyramid Model practices to promote healthy social and emotional development by teaching emotional literacy, positive problem-solving and friendship skills. The model includes three tiers of intervention practice—universal promotion for all children, secondary preventions for children at risk of social-emotional delays, and tertiary interventions for children with persistent challenges. Each tier builds on the preceding tier.

“If we develop a plan around a child with behavior challenges, the strategies that we use are embedded and used throughout the day.”

—m.l. hemmeter.

The Pyramid Model’s first and foundational level is built on nurturing and responsive caregiving for the child. This includes partnerships with families and collaboration among interventionists and classroom teachers. These relationships enable young children to learn healthy social-emotional growth and develop strong, positive relationships with adults and peers.  

“We speak the language all the time,” Cambridge principal DeeAnne Miree says. “It’s not children hearing it just from the classroom teachers; it’s my secretary, it’s the cafeteria and the kitchen staff. Everyone in our building uses positive language with the children. We’re reminding them of our expectations all the time. Those are things that are conducive to helping the students grow and thrive.”

One tool used at Cambridge is “solution cards,” which help children choose positive responses to situations that might otherwise lead to disruptive behavior. For example, if Antonio wants to play with a toy that another child has, he knows he can look at the cards and decide to ask the other child nicely to share the toy, suggest they play together, or offer to trade toys.

If another child is acting out, the cards offer solutions like saying “please stop” or getting a teacher to help. Cambridge students are schooled in positive and productive responses continually.  

“If you have positive teacher-child interactions; clear, consistent, predictable and engaging environments; and good instruction around social-emotional skills, you prevent most problem behavior in pre-K classrooms,” Hemmeter says.

Some children will need more concentrated instruction in identifying and expressing emotions, regulating their responses, solving social problems, initiating and maintaining interactions, responding cooperatively, handling disappointment and anger, and developing friendship skills. This is where the second tier comes in. The child’s teacher might guide and coach families on how to promote targeted social and emotional skills. Families of infants and young toddlers might need support to help their children regulate emotions of stress and understand others’ emotions.  

Each month, Cambridge counselor Ashley Lofties holds sessions for families to teach them how to use positive solutions at home. Maggie Gerges, whose son Kevin was enrolled at the school, says what she learned has changed her relationship with her child.

“I was having a problem with my child not listening or following directions,” Maggie says. “It was very frustrating for me and his dad. I sat down with the counselor, and we spoke about his behavior.”

Working with Lofties, Gerges understood how important routine was for Kevin and how to channel his energy into positive behavior like putting away his toys or going for a walk. Now, she says, Kevin is much better at listening.  

At Cambridge, principal DeeAnne Miree says the Pyramid Model approach is embedded in all communications by teachers and staff members to reinforce positive behaviors cohesively.

Miree credits the Pyramid Model with transforming not only the lives of the children at her school but also her life as an educator and the lives of her teachers and staff.  

“It frees teachers to concentrate on academics instead of behavior issues,” she says.  

Hemmeter oversees training for a $7.7 million Department of Education grant that Metro Nashville Public Schools received to expand the Pyramid Model in pre-K and kindergarten classes district-wide. Along with colleagues from two other universities, she works with MNPS to develop materials and strategies to guide teachers in instituting the model.  

“These projects collectively involve faculty and staff from universities across the country,” Hemmeter says. “They provide an important training opportunity for master’s students here at Peabody who will be future teachers and for doctoral students who will be the next generation of Pyramid Model researchers.”

Through the National Center on Pyramid Model Innovations, Hemmeter supports educators across the country in implementing the Pyramid Model. She and Erin Barton, PhD’07, associate professor of special education at Peabody, are among researchers who provide technical assistance to states as they adopt and scale up sustainable Pyramid Model use within early intervention and early education programs for children from birth to age 5.  

“It frees teachers to concentrate on academics instead of behavior issues.”

—cambridge principal deeanne miree.

Hemmeter also is a co-principal investigator for the first-of-its-kind Infant-Toddler Pyramid Practices project, which will adapt the Pyramid Model for that age range. Research will be carried out in 24 infant-toddler classrooms in Kansas and Tennessee. The Institute of Education Sciences provided a $1.4 million grant to fund the project in collaboration with the University of Kansas.

“We now have research grants testing the Pyramid Model in infant-toddler programs, pre-K programs and kindergarten,” Hemmeter says. “What you do in an infant classroom around Pyramid looks different from what you do in a toddler classroom or in pre-K and kindergarten. It’ll be an interesting test of merging social, emotional and academic skills.”

In three years, the researchers will have collected data around teacher implementation and how well the Pyramid Model works for each age group.  

“I can’t think of another model that has been tested across the whole birth-to-kindergarten range. Most focus more on a specific age range,” Hemmeter says. “We could potentially have a model that is very tightly connected across ages and still protects what’s different about each.”  

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1.2 Hot Buttons Activity

1.2 Milestone Expectations Activity

1.2 Needs Assessment 1: Building Positive Relationships

1.2 Reframing Activity

1.3 Building Positive Relationships Article

1.3 Daily Activities Activity

1.3 Needs Assessment 2 Building Positive Relationships

1.4 Needs Assessment 3 Designing Supportive Environments

1.5 Needs Assessment 4 Designing Supportive Environments

1.6 Needs Assessment 5 Designing Supportive Environments

1.7 Needs Assessment 6 Designing Supportive Environments

1.7 Starters for Positive Feedback

2.2 Needs Assessment 7 Social Emotional Teaching Strategies

2.3 Intentional Instructional Strategies Article

2.3 Needs Assessment 8 Social Emotional Teaching Strategies

2.3 Teaching Targeted Skills Activity

2.4 Embedding Friendship Activity

2.4 Needs Assessment 9 Social Emotional Teaching Strategies

2.4 You've Got to Have Friends Article

2.5 Enhancing Emotional Vocabulary Article

2.5 Needs Assessment 10 Social Emotional Teaching Strategies

2.5 Needs Assessment 11 Social Emotional Teaching Strategies

2.6 Helping Young Children Control Anger and Handle Disappointment Article

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Adult Calm Down Strategies

Behavior Expectations Poster

Breathing Techniques

Calm Down Strategies Poster

Caregiver Planning Form

Early interventionists can use this written manual in the joint planning process with caregivers to organize and write out a plan to implement a caregiving practice with their child in between home visits. Encourage caregivers to leave the plan in an easy...

Caregiver Planning Form (Spanish)

Caregiver Practices to Support Infant-Toddler Social-Emotional Development

This handout outlines the caregiving practices that lead to healthy social-emotional outcomes for infants and toddlers. Early interventionists might use this handout as a resource when engaged in joint planning with caregivers.

Caregiver Responsive Routines Planning Form

Early interventionists can use this form during home visits with caregivers to help them talk through their current daily routines as well as help them to plan out the ways they will embed supports that are responsive to their infant or toddler’s social-e...

Caregiver Responsive Routines Planning Form (Spanish)

Components Behavior Poster

This handout illustrates the components to better understand and respond to behavior in a format that is easy to print.

Development of Play Skills for Infants and Toddlers

This handout provides the developmental milestones of play skills for infant and toddlers.

Family Strategies Poster

Feelings Faces Cards

Infant-Toddler Caregiver Reflection Tool

This tool provides information on the use of caregiver practices to support healthy social emotional development in infants and toddlers. Early interventionists and caregivers can use this tool to build awareness and inform caregivers about what practices...

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Peer Mediated Skills

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Help children learn self-regulation through using the Tucker Turtle Technique.

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Response Strategies when Families Share Hard Things

This tip sheet provides some “in the moment” and “after the visit” response strategies and reflection early intervention providers can use when families share difficult or traumatic experiences.

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Solution Kit

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Solution Kit Home Edition

Solution Kit Instructions Letter

The Developmental Continuum of Social & Emotional Indicators

This one page fact sheet contains the developmental continuum of social and emotional indicators for infant, toddler, and preschool age children.

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