Surface Area or Volume? Prism Word Problems | Grade 8 Math | LevelKite

A prism is a solid with two matching, parallel faces, called its bases, joined by flat side faces. A box is a prism, and so is a wedge with triangle ends. Word problems about prisms usually ask one of two things.

Does something cover the whole outside, like wrapping paper or a fabric cover? Then you need the surface area: the total area of all the faces, counted in square units such as square inches (squares $1$ inch on each side).
Does something fill the inside, like water, sand or foam? Then you need the volume: the amount of space inside, counted in cubic units such as cubic inches (cubes $1$ inch on each edge).

The situation, not the numbers, tells you which one to find.

Why is volume a base area times a length? Say one base has an area of $B$ square units. A layer $1$ unit thick on that base holds $B$ cubic units. Stack layers straight out from the base until you reach the other base. If the bases are $\ell$ units apart, you stack $\ell$ layers, so the volume is $V = B \times \ell$ cubic units. If $\ell$ is not a whole number, the last layer is only part of a layer; a layer $0.5$ unit thick holds $B \times 0.5$ cubic units. So the rule works for any positive length.

Here $\ell$ is the perpendicular length: the distance from one base to the other, measured at a right angle to them. It may be a height, a length or a depth, depending on which faces are the bases. In a box, any two opposite faces can be the bases. Both $B$ and $\ell$ are positive, and they must share one length unit: if $\ell$ is in inches, $B$ is in square inches, and then $V$ is in cubic inches.

Key concept

Covering or filling?

First ask what the situation measures.

Covering the whole outside: find the surface area, the sum of the areas of all the faces in the prism's net (its faces laid out flat). Answer in square units.
Filling the inside: find the volume, the base area $B$ times the perpendicular length $\ell$ . Answer in cubic units.

Write every length with its unit, and make all the units match before you compute.

\text{Surface area} = \text{sum of all face areas} \qquad V = B \times \ell

Box

A

: base length

10

inches, base width

8

inches, height

5

inches. The drawing is not to scale.

Worked example

A gift box

Box $A$ above is a gift box shaped like a rectangular prism, a prism whose six faces are all rectangles. Its base length is $10$ inches, its base width is $8$ inches, and its height is $5$ inches. (a) How much wrapping paper covers the whole box exactly, with no overlap? (b) How much space is inside the box?

Decide what each part measures. In (a), paper lies flat on all six faces, so you need the surface area, in square inches. In (b), the space inside is the volume, in cubic inches. All three lengths are in inches, so the units already match.
Opposite faces of a rectangular prism match, so there are only three different face areas to find, each in square inches.
$10 \times 8 = 80, \quad 10 \times 5 = 50, \quad 8 \times 5 = 40$
Each of those areas appears twice, once for each face in a matching pair. Add two of each to get the surface area.
$2(80) + 2(50) + 2(40) = 160 + 100 + 80 = 340$
For the volume, use the bottom as a base, with an area of 80 square inches. The 5-inch height is perpendicular to the bottom, so it is the perpendicular length. Square inches times inches gives cubic inches.
$V = B \times \ell = 80 \times 5 = 400$

Answer

(a) It takes $340$ square inches of wrapping paper to cover box $A$ . (b) The space inside box $A$ is $400$ cubic inches.

Worked example

A sandbox with mixed units

A sandbox is shaped like a rectangular prism $4$ feet long and $3$ feet wide. How much sand do you need to fill it to a depth of $6$ inches?

Sand fills space, so find the volume. The depth is in inches, but the other lengths are in feet, so change the depth to feet first. Since 12 inches make 1 foot, divide the number of inches by 12.
$6 \text{ inches} = \tfrac{6}{12} \text{ foot} = 0.5 \text{ foot}$
Use the bottom of the sand as a base. Both of its lengths are in feet, so its area is in square feet.
$B = 4 \times 3 = 12$
The depth runs straight down, perpendicular to the bottom, so the perpendicular length is 0.5 foot. Square feet times feet gives cubic feet.
$V = B \times \ell = 12 \times 0.5 = 6$

Answer

You need $6$ cubic feet of sand to fill the sandbox to a depth of $6$ inches.

Net of a triangular prism

The wedge's net: two right triangles with shorter sides

3

and

4

, and three rectangles

10

long and

3

,

4

and

5

wide. Each unit is

1

inch.

Worked example

A foam wedge

A foam wedge that props up a tablet is a prism with right-triangle ends. Each end has legs (the two sides that form the right angle) of $3$ inches and $4$ inches and a longest side of $5$ inches. The wedge is $10$ inches long from one end to the other, and its net is shown above. (a) How much fabric covers the whole wedge exactly? (b) How much foam is inside?

Fabric covers every face, so (a) asks for surface area, in square inches. Foam fills the space, so (b) asks for volume, in cubic inches. Every length is in inches.
Find the area of one triangle. Its legs meet at a right angle, so the 4-inch leg is the height that goes with the 3-inch leg. The slanted 5-inch side is not a height.
$\tfrac{1}{2} \times 3 \times 4 = 6$
In the net, each rectangle is 10 inches long, and its width is one side of the triangle. Find all three rectangle areas.
$3 \times 10 = 30, \quad 4 \times 10 = 40, \quad 5 \times 10 = 50$
Add the areas of all five faces, two triangles and three rectangles, to get the surface area.
$6 + 6 + 30 + 40 + 50 = 132$
For the volume, the bases are the two triangles, because they are the matching, parallel faces, so the base area is 6 square inches. The side faces are rectangles, so the 10-inch length meets each triangle at a right angle; it is the perpendicular length.
$V = B \times \ell = 6 \times 10 = 60$

Answer

(a) It takes $132$ square inches of fabric to cover the wedge. (b) The wedge holds $60$ cubic inches of foam.

Takeaway: The situation decides what you find. If something covers the whole outside, add the areas of all the faces and answer in square units. If something fills the inside, multiply the base area by the perpendicular length and answer in cubic units. In both cases, put every length in the same unit before you compute.