Surface Area of Prisms and Pyramids | Grade 8 Math | LevelKite

The surface area of a prism or pyramid is the total area of all its flat surfaces, called faces. Picture covering every face with paper, with no gaps and no overlaps. The area of that paper is the surface area. Lengths in this lesson are measured in units, so surface areas are in square units. Volume, from the last lesson, measures something different: the cubic units that fill the inside of a solid.

A net is a flat pattern made of all of a solid's faces, joined along some of their edges, that folds up into the solid. Each face appears in the net exactly once, so the total area of the net equals the surface area.

Key concept

Surface area is the sum of every face's area

A prism has two congruent (same size and shape), parallel bases. A pyramid has one base. Every other face is a lateral face. To find surface area, list each face from the net with its dimensions, find each area, and add. A rectangle's area is length times width. A triangle's area is one half times base times height, where the height is perpendicular to that base. Let $S$ be the surface area, $B$ the area of one base, and $L$ the lateral area, which is the total area of the lateral faces. All three are in square units. Then:

\text{Prism: } S = 2B + L \qquad \text{Pyramid: } S = B + L

Net of a triangular prism

The prism's net: two right triangles with sides

3

and

4

forming the right angle, and three rectangles

10

units long that are

3

,

4

and

5

units wide.

Worked example

A triangular prism from its net

Find the surface area of the triangular prism whose net is shown above. Each base is a right triangle. Its legs, the two sides that form the right angle, are $3$ and $4$ units long, and its longest side is $5$ units. The prism is $10$ units long.

List every face from the net with its dimensions, so you do not miss any. The two bases are right triangles with legs 3 and 4. The three lateral faces are rectangles 10 units long. Each rectangle is as wide as the side of the triangle it meets on the prism, so the widths are 3, 4 and 5. That makes 5 faces in all.
Find the area of one base. A triangle's area is one half times base times height, and the height must be perpendicular to the base. The legs meet at the marked right angle, so leg 3 can be the base and leg 4 the height.
$B = \tfrac{1}{2} \times 3 \times 4 = 6$
Each lateral face is a rectangle, so find each area as length times width. Then add the three areas to get the lateral area.
$L = 3 \times 10 + 4 \times 10 + 5 \times 10 = 30 + 40 + 50 = 120$
A prism has two congruent bases, so count the base area twice and then add the lateral area. Every length is in units, so every area is in square units.
$S = 2B + L = 2 \times 6 + 120 = 12 + 120 = 132$

Answer

The surface area of the triangular prism is $132$ square units.

Pyramid $A$ has a square base with sides of $6$ units. Its four triangular faces meet at a point called the apex. It has two different heights:

Its perpendicular height runs from the apex straight down to the base and meets the base at a right angle. It passes through the inside of the solid. It is $4$ units, and it is the kind of height you used for volume.
Its slant height is the height of each triangular face. It runs along a face from the middle of a base edge up to the apex and meets that edge at a right angle. It is $5$ units.

Each triangular face uses the slant height, not the perpendicular height. The drawing of pyramid $A$ below marks its base length and its perpendicular height. The net after the drawing shows the slant height.

Pyramid

A

, not drawn to scale. Its base length is

6

units and its perpendicular height is

4

units. Its slant height,

5

units, is on the net.

Net of a square pyramid

The net of pyramid

A

: a square base

6

units on a side and four triangles, each with base

6

and slant height

5

.

Worked example

A square pyramid from its net

Find the surface area of pyramid $A$ from its net. Its base is a square with sides of $6$ units, its perpendicular height is $4$ units, and its slant height is $5$ units.

List every face from the net with its dimensions. There is one square base, 6 units by 6 units. There are four triangular lateral faces, each with a base of 6 units and a slant height of 5 units. That makes 5 faces.
Find the base area. The base is a square, so multiply its length by its width.
$B = 6 \times 6 = 36$
Find the area of one triangular face with one half times base times height. Use the slant height, 5, as the height, because it lies on the face and meets the base edge at a right angle. The perpendicular height, 4, does not lie on any face, so you do not use it here.
$\tfrac{1}{2} \times 6 \times 5 = 15$
Each of the four triangles has base 6 and slant height 5, so each has an area of 15. Multiply by 4 to get the lateral area.
$L = 4 \times 15 = 60$
A pyramid has only one base, so add the base area once to the lateral area. Both areas are in square units, so the sum is in square units too.
$S = B + L = 36 + 60 = 96$

Answer

The surface area of pyramid $A$ is $96$ square units.

Takeaway: Surface area is the total area of every face, in square units. Use the net to list each face with its dimensions, find each area, and add. Count both bases of a prism, the single base of a pyramid, and every lateral face. For a pyramid's triangular faces, use the slant height, which is the height of each face, and not the perpendicular height. Next, you will use surface area and volume to solve word problems.