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Pyramid

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Pyramid

Study Guide

Key Definition

A pyramid is a polyhedron with a polygonal base and triangular faces that meet at a point called an apex. The lateral surface area of a regular pyramid is calculated using $L.S.A. = \frac{1}{2} p l$, where $p$ is the perimeter of the base, and $l$ is the slant height.

Important Notes

A regular pyramid has a base that is a regular polygon.
The lateral faces of a regular pyramid are congruent isosceles triangles.
The altitude of a regular pyramid meets the base at its center.
Slant height is only defined for regular pyramids.
The total surface area includes the base and the lateral surface area.
The volume of a pyramid is given by V = 1/3 B h.

Mathematical Notation

$\frac{1}{2}$ means half

$l$ is slant height

$p$ is perimeter of the base

$h$ is the altitude (height) of the pyramid

$B$ is area of the base

$V$ is volume of the pyramid

Remember to use proper notation when solving problems

Why It Works

Each lateral face is an isosceles triangle with base equal to one side of the base polygon and height equal to the slant height l. The area of one triangle is (1/2)*side*l, and summing over all sides yields (1/2)*p*l.

Remember

To find the lateral surface area of a regular pyramid, use $L.S.A. = \frac{1}{2} p l$. To find the volume, use $V = \frac{1}{3} B h$.

Quick Reference

Lateral Surface Area:$L.S.A. = \frac{1}{2} p l$

Total Surface Area:$T.S.A. = \frac{1}{2} p l + B$

Volume:$V = \frac{1}{3} B h$

Understanding Pyramids

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Video explanation of this concept

Beginner

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Beginner Explanation

A regular pyramid has a polygonal base and congruent triangular lateral faces. Each triangle has base equal to one side of the base and slant height l, so its area is $(1/2) s l$. Summing over all sides gives $L.S.A. = \frac{1}{2} p l$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the lateral surface area of a regular pyramid with a triangular base with each edge measuring 8 inches and a slant height of 5 inches? Use $L.S.A. = \frac{1}{2} p l$.

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A teenager wants to build a model pyramid with a square base of side 10 cm and slant height 7 cm. Find the lateral surface area using $L.S.A. = \frac{1}{2} p l$.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Imagine a pyramid with a hexagonal base. How would you calculate the lateral surface area if each side of the base is 6 cm and the slant height is 10 cm?

Challenge Quiz

Single Choice Quiz

Advanced

A regular pyramid with a pentagonal base has a side length of 4 cm and a slant height of 9 cm. What is its lateral surface area?

Recap

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Review key concepts and takeaways