Develop the surface-area relationship for a pyramid

Ontario Grade 10 Mathematics

Measurement and Trigonometry

Build the relationship by adding the base and triangular faces

A pyramid has one base and triangular side faces that meet at a point called the apex. Its surface area is the total area of all its outside faces. In this lesson, you will build a relationship for a regular pyramid. Its base is a regular polygon, and its side triangles have the same slant height. The relationship shows how the base perimeter and slant height combine to give the area of the triangular faces.

What you will learn

Identify the faces that make up a pyramid’s surface area.
Distinguish the base area from the area of the triangular side faces.
Develop and use the surface-area relationship for a regular pyramid.
Choose measurements that match the triangle heights.

1. Start with familiar area ideas

You already know how to find the area of a rectangle: multiply its length by its width. For a triangle, multiply its base by its perpendicular height, then divide by two. Perpendicular lines meet at a right angle.

A pyramid’s outside is made of its base and triangular side faces. To find its surface area, find the area of each face and add the areas together.

The height of a triangular face is measured at a right angle to that triangle’s base. In a regular pyramid, this face height is called the slant height. It runs along a side face from the apex to the midpoint of a base edge. It is not the vertical height inside the pyramid.

A_{\triangle}=\frac{1}{2}bh

Surface area is the sum of the areas of all outside faces.
Use the perpendicular height of a triangle to find its area.
The slant height belongs to a triangular side face.

2. Combine the triangular faces

Imagine unfolding the pyramid so that its faces lie flat. This flat layout is called a net. It shows one base and all the triangular side faces.

Each triangle’s base matches an edge of the pyramid’s base. Its area is half its base-edge length multiplied by the slant height. In a regular pyramid, all the side triangles have the same slant height.

Add the areas of the side triangles. Their base-edge lengths add up to the perimeter of the base. Perimeter means the total distance around a shape. So the combined area of the triangles is half the base perimeter multiplied by the slant height. This combined area is called the lateral area.

The total surface area includes the base as well as the side triangles. Add the base area to the lateral area. In the relationship,

B

means the base area,

P

means the base perimeter, and

\ell

means the slant height.

SA=B+\frac{1}{2}P\ell

A net helps show which faces need to be included.
The triangle bases together make the base perimeter.
Lateral area is the area of the triangular side faces only.
Surface area equals base area plus lateral area.

3. Match each measurement to the correct part

The relationship applies to a regular pyramid: one with a regular polygon base and equal slant heights on its side triangles. A regular polygon has equal sides and equal angles.

Check what each measurement describes before using it. Find the base area from the shape at the bottom. Find the perimeter by adding the base’s edge lengths. Use the slant height of a triangular side face in the lateral-area part.

If a problem gives the pyramid’s vertical height, do not substitute it for slant height. They measure different distances. The vertical height is inside the pyramid; slant height lies on a triangular face.

Use consistent units for all lengths. Give area in square units, such as square centimetres. For a square base with side length

s

, its perimeter is four times

s

, and its area is

s

multiplied by itself.

P=4s,\quad B=s^2

Use the base area and base perimeter in their proper roles.
Use the side triangle’s slant height, not the pyramid’s vertical height.
Write the final surface area in square units.

4. Guided example and independent practice

In the example, find the base area and perimeter first. Then find the area of the triangular faces. Add the base area last. This keeps the two parts of the total separate.

For independent practice, consider a regular square pyramid with a base side length of

7

cm and a slant height of

9

cm. Find the base area, perimeter, lateral area, and total surface area. Check that the lateral area includes only the triangles and that your final answer uses square centimetres.

Find the base area and perimeter before the lateral area.
Add the base area exactly once.
Check that your area units are squared.

Parts of a regular pyramid’s surface area

Part	How to find its area	How it contributes
Base	Find the area of the base shape.	Add the base area once.
Triangular side faces	Multiply half the base perimeter by the slant height.	This is the lateral area.
Whole outside	Add the base area and lateral area.	This is the surface area.

Worked example

A regular square pyramid

A regular square pyramid has a base side length of

6

cm and a slant height of

8

cm. Find its surface area.

Find the base area
The base is a square. Multiply its side length by itself to find the area.
$B=6\times6=36\text{ cm}^2$
Find the base perimeter
A square has four equal sides. Multiply the side length by four to find the total distance around the base.
$P=4\times6=24\text{ cm}$
Find the lateral area
Multiply half the perimeter by the slant height. This gives the combined area of the four triangular faces.
$\frac{1}{2}P\ell=\frac{1}{2}\times24\times8=96\text{ cm}^2$
Add the base area
Surface area includes the base and the triangular faces. Add their areas to find the total.
$SA=36+96=132\text{ cm}^2$

Answer: The surface area is

132\text{ cm}^2

.

Check: The base area is

36\text{ cm}^2

and the lateral area is

96\text{ cm}^2

. Their sum is

132\text{ cm}^2

, which includes the square base and all four triangular faces.

Common mistakes and how to avoid them

Using the pyramid’s vertical height in the lateral-area relationship.

Correction: Use the slant height measured on a triangular face, at a right angle to that face’s base edge.

Calling the lateral area the total surface area.

Correction: Lateral area includes only the side faces. Add the base area to get the total surface area.

Using the base area instead of the base perimeter in the lateral-area calculation.

Correction: The relationship uses the total length around the base, not the area inside it.

Giving an area in ordinary length units, such as centimetres.

Correction: Area uses square units, such as

\text{cm}^2

.

Lesson summary

A pyramid’s surface area includes its base and its triangular side faces.
For a regular pyramid, the triangular faces together have an area equal to half the base perimeter multiplied by the slant height.
Add the base area to the lateral area to find the total surface area.
In the relationship, $B$ is base area, $P$ is base perimeter, and $\ell$ is slant height.
Use square units for the final area.

Check your understanding

Question 1

A regular square pyramid has a base side length of

5

cm and a slant height of

6

cm. What is its surface area?

$85\text{ cm}^2$
$60\text{ cm}^2$
$95\text{ cm}^2$
$30\text{ cm}^2$

Show answer and explanation

85\text{ cm}^2

The square base area is

25\text{ cm}^2

, and the perimeter is

20

cm. The lateral area is half of

20

multiplied by

6

, or

60\text{ cm}^2

. Add the base area:

25+60=85\text{ cm}^2

.

Question 2

Which measurement is used with the base perimeter to find the lateral area of a regular pyramid?

The area of the base
The slant height of a side triangle
The length of one base edge only
The vertical height of the pyramid

Show answer and explanation

The slant height of a side triangle

The triangular face areas use the height measured on the side faces. For a regular pyramid, this shared face height is the slant height.

Key terms

Apex: The point where the triangular side faces of a pyramid meet.
Base: The polygon at the bottom of a pyramid.
Base perimeter: The total length around the base.
Lateral area: The combined area of a pyramid’s triangular side faces.
Net: A flat layout that shows the faces of a three-dimensional shape.
Regular pyramid: A pyramid with a regular polygon base and equal slant heights on its triangular side faces.
Slant height: The perpendicular height of a triangular side face, measured from the apex to a base edge.
Surface area: The total area of all the outside faces of a solid.

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