Surface Area and Volume of Prisms: Solve and Check | Grade 7 Math | LevelKite

Think about a solid wooden block. You can ask two different questions about it.

How much wood is in it? That is its volume: the amount of space a solid fills.
How much surface would you paint to cover it? That is its surface area: the total area of all its faces, the flat surfaces on the outside of the solid.

You already know how to find a volume and how to find a missing length from a volume. In this lesson you will find both measures for one solid. Then you will read other people's volume work, find the mistakes, and fix them.

Key concept

Volume fills a prism; surface area covers it

These rules hold for a right prism when every measure is positive and all lengths use one unit.

Volume $V$ is the space the prism fills. Let $B$ be the base area, the area of one base, and let $h$ be the height, the perpendicular distance between the bases. A layer $1$ unit tall on the base fills as many cubic units as the base has square units, and $h$ tells how many of those layers fit between the bases. So $V = B \times h$ .

Surface area $SA$ is the sum of the areas of all the faces.

Lengths use linear units such as $\text{cm}$ . Areas use square units such as $\text{cm}^2$ . Volumes use cubic units such as $\text{cm}^3$ . A height is always perpendicular to its base. A slanted length, one that is not perpendicular to the base, is not a height.

V = B \times h

Length, area and volume when lengths are in centimeters

Measure	What it tells you	Kind of unit	Where the unit comes from
Length	How long a side or a height is	Linear: $\text{cm}$	A single length
Area of one face	How much flat surface that face covers	Square: $\text{cm}^2$	Length times length: $\text{cm} \times \text{cm}$
Surface area	How much flat surface all the faces cover together	Square: $\text{cm}^2$	Adding areas keeps $\text{cm}^2$
Volume	How much space the solid fills	Cubic: $\text{cm}^3$	Area times length: $\text{cm}^2 \times \text{cm}$

Some problems ask about both measures of one prism. Before you calculate, write down the base shape, the base dimensions (the lengths that give the base its size), the perpendicular height of the prism, and the unit. Then decide which measure each question needs.

A question about how much space the solid fills, or how much it holds, needs volume.
A question about covering the whole outside, such as wrapping or painting every face, needs surface area.

The two measures use the same lengths in different ways. So you work out each measure separately and give each one its own kind of unit: cubic units for volume and square units for surface area.

Worked example

One block, two measures

A solid wooden block is a right rectangular prism. Its base is a rectangle $10\text{ cm}$ long and $4\text{ cm}$ wide. The height of the prism, perpendicular to the base, is $3\text{ cm}$ . How much wood is in the block? How much surface do you cover if you paint all six faces?

The amount of wood is the space the block fills, so you need the volume. Start with the base area B. The base is a rectangle, so multiply its length by its width. Centimeters times centimeters gives square centimeters.
$B = 10 \times 4 = 40\text{ cm}^2$
Volume is the base area times the perpendicular height, so multiply 40 square centimeters by the height h, which is 3 cm. Square centimeters times centimeters gives cubic centimeters, the kind of unit a volume needs.
$V = B \times h = 40 \times 3 = 120\text{ cm}^3$
Paint covers the faces, so you need the surface area SA. The six faces come in three matching pairs: the two bases and two pairs of side faces. Because this is a right prism, each side face is a rectangle with one side of the base as one dimension and the height, 3 cm, as the other. Find one area from each pair.
$10 \times 4 = 40\text{ cm}^2, \quad 10 \times 3 = 30\text{ cm}^2, \quad 4 \times 3 = 12\text{ cm}^2$
Surface area is the sum of all six face areas. Each area belongs to two faces, so double each one, then add. The total is still an area, in square centimeters.
$SA = 2(40) + 2(30) + 2(12) = 80 + 60 + 24 = 164\text{ cm}^2$
Match each result to its question. The wood is a volume, in cubic centimeters. The painted surface is an area, in square centimeters. These are different kinds of measure, so you cannot say that one is larger than the other.

Answer

The block contains $120\text{ cm}^3$ of wood, and painting all six faces covers $164\text{ cm}^2$ of surface.

Worked example

From a volume to a surface area

A solid wedge is a right triangular prism with a volume of $540\text{ cm}^3$ . Its base is a triangle with sides $9\text{ cm}$ , $12\text{ cm}$ and $15\text{ cm}$ . The $9\text{ cm}$ and $12\text{ cm}$ sides meet at a right angle. Find the height of the prism, which is the perpendicular distance between the two triangular bases. Then find the surface area.

Find the base area B. The area of a triangle is half of one side times the height perpendicular to that side. The 9 cm and 12 cm sides are perpendicular, so use one as the side and the other as its height. The 15 cm side is slanted. It is not perpendicular to either of them, so it is not a height here.
$B = \tfrac{1}{2} \times 9 \times 12 = \tfrac{1}{2} \times 108 = 54\text{ cm}^2$
The problem gives the volume and leaves out the height, so work backward as you have done before. Let h be the height of the prism in centimeters. Put the known values into the volume relationship, V equals B times h.
$540 = 54 \times h$
Divide both sides by 54. Dividing both sides of an equation by the same number keeps it true as long as that number is not zero, and 54 is positive. Cubic centimeters divided by square centimeters gives centimeters, and the result is positive, as a length must be.
$h = 540 \div 54 = 10\text{ cm}$
Now you know every length, so list the five faces. Two are the triangular bases, 54 square centimeters each. Because this is a right prism, the other three are rectangles. Each one has a side of the triangle as one dimension and the height, 10 cm, as the other.
$9 \times 10 = 90\text{ cm}^2, \quad 12 \times 10 = 120\text{ cm}^2, \quad 15 \times 10 = 150\text{ cm}^2$
Surface area is the sum of all the face areas, so add the five areas. You use the slanted 15 cm side here because it is a side of one of the rectangular faces. You did not use it for the volume because it is not a height.
$SA = 54 + 54 + 90 + 120 + 150 = 108 + 360 = 468\text{ cm}^2$
Check the height. Put 10 back in for h and multiply by the base area. You get the volume that the problem gave, so the height is right.
$54 \times 10 = 540$

Answer

The height of the prism is $10\text{ cm}$ , and its surface area is $468\text{ cm}^2$ .

Same volume, same surface area? Compare two right rectangular prisms

Prism	Base	Perpendicular height	Volume	Surface area
A	Rectangle, $4\text{ cm}$ by $4\text{ cm}$	$4\text{ cm}$	$(4 \times 4) \times 4 = 64\text{ cm}^3$	$2(16) + 2(16) + 2(16) = 96\text{ cm}^2$
B	Rectangle, $8\text{ cm}$ by $4\text{ cm}$	$2\text{ cm}$	$(8 \times 4) \times 2 = 64\text{ cm}^3$	$2(32) + 2(16) + 2(8) = 112\text{ cm}^2$

Error analysis means reading a finished solution, finding the step that goes wrong, explaining why it is wrong, and then correcting it. You are not only checking the final number. You are checking that every step matches what volume means.

For the volume of a right prism, with positive measures in one unit, ask four questions in order.

Measure: Does the work find the space the prism fills? Adding face areas finds surface area instead.
Base area: Is $B$ the area of one base, found with the area rule for that shape?
Height: Is each height perpendicular to its base? Check the height inside a triangular base and the height of the prism. A slanted length is not a height.
Units: Are all the lengths in one unit, and is the answer in cubic units?

Try three of the checks on the wedge from the second example. It is a right triangular prism. Its base is a triangle whose $9\text{ cm}$ and $12\text{ cm}$ sides are perpendicular, so the base area is $54\text{ cm}^2$ . The height of the prism, perpendicular to the base, is $10\text{ cm}$ , so the volume is $54 \times 10 = 540\text{ cm}^3$ .

Each row of the table gives a different attempt at that volume. Before you read across a row, decide which check the work fails.

Three attempts at the volume of the wedge

Student's work	Check it fails	Why it fails	Corrected work
$V = 54 + 54 + 90 + 120 + 150 = 468$	Measure	Adding the five face areas gives the surface area, $468\text{ cm}^2$ , not the space filled.	$V = 54 \times 10 = 540\text{ cm}^3$
$V = 9 \times 12 \times 10 = 1{,}080\text{ cm}^3$	Base area	$9 \times 12$ is the area of a rectangle. The triangular base is half of that rectangle.	$V = \tfrac{1}{2} \times 9 \times 12 \times 10 = 540\text{ cm}^3$
$V = \tfrac{1}{2} \times 9 \times 12 \times 10 = 540\text{ cm}^2$	Units	The number is right, but $\text{cm}^2$ measures area. A volume needs cubic units.	$V = 540\text{ cm}^3$

Worked example

Find and fix the mistake

A right triangular prism has a triangle for its base. One side of the triangle is $10\text{ cm}$ , the height perpendicular to that side is $12\text{ cm}$ , and the other two sides are $13\text{ cm}$ each. The height of the prism, perpendicular to the base, is $20\text{ cm}$ . Kai writes $B = \tfrac{1}{2} \times 10 \times 13 = 65\text{ cm}^2$ and then $V = 65 \times 20 = 1{,}300\text{ cm}^3$ . Find Kai's mistake and correct it.

Check the measure. Kai multiplies a base area by the height of the prism. That is base area times height, so Kai is finding the space the prism fills, as the problem asks. This check passes.
$V = B \times h$
Check the base area and the height inside it. The area of a triangle is half of one side times the height perpendicular to that side. Kai has the half and the 10 cm side, but pairs them with 13 cm. That is a slanted side, not perpendicular to the 10 cm side, so it is not the height. The perpendicular height is 12 cm. Here is the mistake.
Check the height of the prism and the units. The height, 20 cm, is perpendicular to the base, so Kai uses it correctly. Every length is in centimeters, and Kai gives the volume in cubic centimeters. These checks pass.
Correct the base area. The rule needs the height perpendicular to the 10 cm side, so use 12 cm in place of 13 cm.
$B = \tfrac{1}{2} \times 10 \times 12 = \tfrac{1}{2} \times 120 = 60\text{ cm}^2$
Volume is the base area times the perpendicular height, so multiply the corrected base area by 20 cm. Square centimeters times centimeters gives cubic centimeters.
$V = 60 \times 20 = 1{,}200\text{ cm}^3$
Say what the mistake did. Here the slanted side, 13 cm, is longer than the perpendicular height, 12 cm. That made Kai's base area and volume too large.
$1{,}300 - 1{,}200 = 100\text{ cm}^3$

Answer

Kai used the slanted $13\text{ cm}$ side as the height of the triangular base. With the perpendicular height, $12\text{ cm}$ , the base area is $60\text{ cm}^2$ and the volume is $1{,}200\text{ cm}^3$ .

Here is the idea to keep. Volume and surface area describe one prism in two different ways. For a right prism with positive measures in one unit, the volume is the space it fills: the base area times the perpendicular height, in cubic units. The surface area is its cover: the sum of the areas of all its faces, in square units. Work out each measure on its own, because prisms with the same volume can have different surface areas.

When you check volume work, ask four questions. Is it the right measure? Is the base area right for the base shape? Is every height perpendicular, not slanted? Are the lengths in one unit and the answer in cubic units? When a step fails a check, say why it fails, and then redo the work from that step. The same four questions work on your own solutions too.