Surface Area of Triangular Prisms and Pyramids | Grade 7 Math | LevelKite

You already know how to find the surface area of a rectangular prism: find the area of each of its six rectangular faces, then add them. Surface area is the total area of all the faces of a solid. The faces are its flat outside surfaces.

That idea stays the same in this lesson. What changes is the shape of the faces: now some of them are triangles. You will work with two solids, triangular prisms and pyramids. For each one you will list every face with its own measurements, find each area, and add. The faces do not overlap, and together they cover the whole outside, so the sum counts every part of the surface exactly once.

Two conditions hold in every problem here. Every length is a positive number, and all the lengths in one problem use the same unit. Lengths are in linear units such as cm. Areas are in square units such as $\text{cm}^2$ . Cubic units such as $\text{cm}^3$ measure volume, which is a different quantity, so they never label a surface area.

Every face in this lesson is a rectangle or a triangle, so two area rules do all the work.

Rectangle: area is length times width.
Triangle: area is half of base times height.

For a triangle, the base can be any one of its sides. The height is the distance from that side to the opposite corner, measured perpendicular to the base, which means at a right angle to it. A side of the triangle is the height only when it meets the base at a right angle, as in a right triangle.

Why half? Cut a rectangle from one corner to the opposite corner and you get two equal right triangles, so each one has half of the rectangle's area. The same half rule works for every triangle, as long as the height is perpendicular to the base.

Both rules need positive lengths in the same unit, and they give an area in square units. Here is the triangle rule in symbols, where $b$ is the length of the base, $h$ is the height perpendicular to that base, and $A$ is the area.

A = \frac{1}{2} \times b \times h

A triangular prism is a solid with two identical triangles at opposite ends, joined by flat side faces. The two triangles are its bases. The side faces are its lateral faces, because lateral means on the side.

Every prism in this lesson is a right prism. That means the edges joining the two bases meet them at right angles, so each lateral face is a rectangle. (An edge is a line segment where two faces meet.) A triangle has three sides, and each side is an edge of one rectangle. So a right triangular prism has $2 + 3 = 5$ faces: two triangular bases and three rectangles.

Each rectangle has two measurements. One is a side of the triangle. The other is the length of the prism, which is the perpendicular distance between the two bases.

One caution: the word base has two meanings. A base of the prism is a whole triangular face. The base of a triangle is one of its sides.

Faces of a right triangular prism (positive lengths in one unit, areas in square units)

Face	How many	Shape	Measurements it needs	Area of one face
Base	2	Triangle	One side of the triangle (its base) and the height perpendicular to that side	Half of base times height
Lateral face	3	Rectangle	One side of the triangle and the length of the prism	Side of triangle times length of prism

Worked example

Surface area of a triangular prism

A right triangular prism has two triangular bases. Each base is a triangle with sides of $13$ cm, $14$ cm and $15$ cm. The height of the triangle perpendicular to its $14$ cm side is $12$ cm. The length of the prism is $10$ cm, so its three rectangular lateral faces measure $13$ cm by $10$ cm, $14$ cm by $10$ cm and $15$ cm by $10$ cm. Find the surface area of the prism.

List every face with its measurements so that none is missed. There are two triangular bases, each with base 14 cm and height 12 cm. There are three rectangular lateral faces, one for each side of the triangle: 13 cm by 10 cm, 14 cm by 10 cm and 15 cm by 10 cm.
$2 \text{ triangles} + 3 \text{ rectangles} = 5 \text{ faces}$
Find the area of one triangular base. The 12 cm height is perpendicular to the 14 cm side, so those two measurements go together. The 13 cm and 15 cm sides do not meet the 14 cm side at a right angle, so neither one is the height. Centimeters times centimeters gives square centimeters.
$\tfrac{1}{2} \times 14 \times 12 = \tfrac{1}{2} \times 168 = 84$
The two bases have the same measurements, so they have the same area. Adding that area twice is the same as multiplying it by 2.
$2 \times 84 = 168$
Find the area of each rectangle with length times width. One measurement is a side of the triangle and the other is the 10 cm length of the prism. The 12 cm height of the triangle is not an edge of any rectangle, so it is not used here.
$13 \times 10 = 130, \quad 14 \times 10 = 140, \quad 15 \times 10 = 150$
Add the three rectangle areas to get the total area of the lateral faces.
$130 + 140 + 150 = 420$
Add the area of the two bases and the area of the three lateral faces, so that all five faces are counted once. Every area is in square centimeters, so the total is too.
$168 + 420 = 588$

Answer

The surface area of the triangular prism is $588$ square centimeters, or $588\ \text{cm}^2$ .

Now for the second solid. A pyramid has one flat base and triangular lateral faces that all meet at one point, called the apex. Each side of the base is the bottom edge of one triangle, so there is one triangle for each side of the base. A pyramid with a square base has $1 + 4 = 5$ faces.

To find the area of a triangular face you need its base and its height. Its base is a side of the pyramid's base. Its height has a special name, the slant height: the distance from the apex down to that face's base edge, measured along the face and perpendicular to that edge. Each triangular face has its own slant height.

A pyramid also has a height of the pyramid: the distance from the apex straight down to the base, perpendicular to the base. In this lesson the apex is always above the center of the base, so this height passes through the inside of the solid and does not run along any face.

These are two different measurements. Of the two, only the slant height is the height of a face, so it is the one you use for surface area.

Two different measurements of a pyramid whose apex is above the center of its base

Measurement	Where it is	Perpendicular to	Used for surface area?
Height of the pyramid	Inside the solid, from the apex straight down to the base	The base of the pyramid	No. It is not the height of any face.
Slant height of a lateral face	On the outside, along one triangular face, from the apex to that face's base edge	The base edge of that triangular face	Yes. It is the height of that triangle.

Key concept

List every face, then add the areas

The surface area of a triangular prism or a pyramid is the sum of the areas of all its faces: its base or bases plus every lateral face.

A right triangular prism has 2 triangular bases and 3 rectangles.
A pyramid has 1 base and one triangle for each side of that base.

Match each face with its own measurements. A triangle's height must be perpendicular to the side you use as its base. For a lateral face of a pyramid, that height is the face's slant height, not the height of the pyramid.

This works when every length is positive and all lengths use one unit. The answer is then in square units.

\text{surface area} = \text{area of base(s)} + \text{area of all lateral faces}

Worked example

Surface area of a square pyramid

A pyramid has a square base with sides of $6$ cm, and its apex is above the center of the base. Each of its four triangular lateral faces has a base of $6$ cm and a slant height of $5$ cm. The height of the pyramid is $4$ cm. Find the surface area of the pyramid.

List every face with its measurements. There is one square base, 6 cm by 6 cm. There are four triangular lateral faces, each with base 6 cm and slant height 5 cm. The 4 cm height of the pyramid is inside the solid. It is not the height of any face, so it is not used.
$1 \text{ square} + 4 \text{ triangles} = 5 \text{ faces}$
Find the area of the square base with length times width. Centimeters times centimeters gives square centimeters.
$6 \times 6 = 36$
Find the area of one triangular face. Its slant height is perpendicular to its 6 cm base edge, so the slant height is the height of the triangle.
$\tfrac{1}{2} \times 6 \times 5 = \tfrac{1}{2} \times 30 = 15$
The four triangles have the same base and the same slant height, so they have the same area. Multiply the area of one triangle by 4.
$4 \times 15 = 60$
Add the area of the base and the area of the four lateral faces. Every area is in square centimeters, so the total is too.
$36 + 60 = 96$

Answer

The surface area of the square pyramid is $96$ square centimeters, or $96\ \text{cm}^2$ .

Worked example

A pyramid with two different slant heights

A pyramid has a rectangular base that is $30$ m long and $12$ m wide, and its apex is above the center of the base. Two of its triangular lateral faces each have a base of $30$ m and a slant height of $10$ m. The other two each have a base of $12$ m and a slant height of $17$ m. The height of the pyramid is $8$ m. Find the surface area of the pyramid.

List the faces: one rectangular base, 30 m by 12 m; two triangles with base 30 m and slant height 10 m; and two triangles with base 12 m and slant height 17 m. The 8 m height of the pyramid is not the height of any face, so leave it out.
$1 + 2 + 2 = 5 \text{ faces}$
Find the area of the rectangular base with length times width.
$30 \times 12 = 360$
Find the area of one triangle on a 30 m edge, using its own slant height of 10 m. Then double it, because there are two of these triangles.
$\tfrac{1}{2} \times 30 \times 10 = 15 \times 10 = 150, \quad 2 \times 150 = 300$
Find the area of one triangle on a 12 m edge. Its slant height is 17 m, not 10 m. Then double it.
$\tfrac{1}{2} \times 12 \times 17 = 6 \times 17 = 102, \quad 2 \times 102 = 204$
Add the areas of all five faces. The lengths were in meters, so every area and the total are in square meters.
$360 + 300 + 204 = 864$

Answer

The surface area of the pyramid is $864$ square meters, or $864\ \text{m}^2$ .

Takeaway: Surface area is the sum of the areas of all the faces of a solid. For a right triangular prism, add two triangular bases and three rectangles. For a pyramid, add one base and one triangle for each side of the base, using each triangle's slant height and not the height of the pyramid. Keep every length positive and in one unit, pair each triangle's base only with the height perpendicular to it, and write the answer in square units.

Next, you will use this method to solve surface area word problems and to find and fix errors in surface area work.