Missing Dimensions and Volume Word Problems | Grade 7 Math | LevelKite

In the last lesson you found the volume of a right prism, the amount of space it fills, from its measurements. Now you will work backward: the volume is known, and one dimension is missing. A dimension is a length that describes the solid, such as a side of its base or its height.

A right prism is a solid with two parallel faces of the same shape and size, called bases, joined by rectangles that meet the bases at right angles. Think of filling a right prism from one base to the other. Every slice you add covers the same base, so the space filled depends on only two things: how much area the base covers, and how far apart the two bases are. Once you see volume as built from those two parts, you can undo the multiplication to find a missing part.

Key concept

Volume is base area times height

For a right prism, let $B$ be the base area: the area of one base, in square units such as $\text{cm}^2$ . Let $h$ be the height, in linear units (units of length) such as cm. Let $V$ be the volume, in cubic units such as $\text{cm}^3$ . The height is the perpendicular distance between the two bases: it is measured at a right angle to them and is never a slanted length. All three measures are positive, and all are built from the same unit of length.

Every 1 unit of height adds $B$ cubic units of space, so $h$ units of height give $B \cdot h$ cubic units. If you know the volume and one of the two numbers being multiplied, the missing one is the number that makes the product true.

V = B \cdot h

Three kinds of measure in a right prism

Kind of measure	What it measures	Unit when lengths are in cm	Where it appears
Linear (length)	Distance in one direction	cm	Height $h$ and each side of a base
Square (area)	Flat surface covered	$\text{cm}^2$	Base area $B$
Cubic (volume)	Space filled	$\text{cm}^3$	Volume $V$

Suppose you know $V$ and $B$ but not $h$ . The equation $V = B \cdot h$ says that $h$ is the number you multiply by $B$ to get $V$ . Division answers exactly that question.

Dividing both sides of an equation by the same nonzero number keeps the two sides equal. The number you divide by may be a whole number, a fraction or a decimal; it only has to be nonzero. A base area is positive, so $B$ is never zero, and you may divide both sides by $B$ . A height is positive too, so when $B$ is the missing part you may divide both sides by $h$ .

The units agree. When every measure is built from the same unit of length, cubic units divided by square units leave linear units, so $V \div B$ is a length. Cubic units divided by linear units leave square units, so $V \div h$ is an area. Written as equations, the two undoing steps are:

h = \frac{V}{B} \qquad \text{and} \qquad B = \frac{V}{h} \qquad (B > 0,\ h > 0)

Worked example

Missing height of a rectangular prism

A right rectangular prism has volume $V = 360\text{ cm}^3$ . Its base is a rectangle $9$ cm long and $5$ cm wide. Find the prism's height $h$ in cm, the perpendicular distance between its two bases.

Find the base area B. The base is a rectangle, so multiply its length by its width. Two lengths in cm multiply to give an area in square centimeters.
$B = 9 \cdot 5 = 45 \text{ cm}^2$
Write the relationship with the values you know: for a right prism, volume equals base area times perpendicular height. The height h is the only unknown.
$360 = 45 \cdot h$
Divide both sides by 45 to undo the multiplication. The two sides stay equal because 45 is not zero, and h now stands alone. Since 45 times 8 is 320 plus 40, or 360, the result is 8.
$h = \frac{360}{45} = 8$
Name the unit and check. Cubic centimeters divided by square centimeters give centimeters, so h is 8 cm. Multiplying the base area by this height must return the given volume.
$45 \cdot 8 = 360 \text{ cm}^3$

Answer

The height of the prism is $8$ cm.

Sometimes the missing dimension is a side of the base, not the prism's height. Then you undo the volume in two stages. First find the base area with $B = V \div h$ , which is allowed because a height is positive and so is not zero. Then use the area rule for the base shape to find the missing side. For a rectangular base, the area is length times width, so a missing width is the base area divided by the length. That division is allowed too, because a length is positive.

A triangular prism needs extra care, because the word height can appear twice. The prism height is the perpendicular distance between the two triangular bases. The triangle height is the perpendicular distance from one side of the triangle to the opposite corner. Neither one is measured along a slanted side. Give each height its own letter so you always know which one an equation is about.

Worked example

Missing height of a triangular base

A right triangular prism has volume $V = 420\text{ cm}^3$ . Its prism height is $h = 14$ cm, measured perpendicular to the two triangular bases. Each base is a triangle with one side $8$ cm long. Find the triangle height $a$ in cm, the perpendicular distance from that $8$ cm side to the opposite corner.

Because this is a right prism with all lengths in cm, write volume equals base area times prism height with the values you know. The base area B is not known yet, so it stays as a letter.
$420 = B \cdot 14$
Divide both sides by 14, which keeps them equal because 14 is not zero. Since 14 times 30 is 420, the result is 30. Cubic centimeters divided by centimeters give square centimeters, so this is an area.
$B = \frac{420}{14} = 30 \text{ cm}^2$
Connect B to the unknown triangle height a. The area of a triangle is one half of a side times the height perpendicular to that side.
$30 = \tfrac{1}{2} \cdot 8 \cdot a$
Multiplying one half by 8 first does not change the product, and it gives 4. Now the right side is a single number times a.
$30 = 4 \cdot a$
Divide both sides by 4, which is not zero, so that a stands alone. Four goes into 30 seven times with 2 left over, and 2 divided by 4 is 0.5. Square centimeters divided by centimeters give centimeters.
$a = \frac{30}{4} = 7.5 \text{ cm}$
Check by rebuilding the volume: first the triangle's area, then that area times the prism height.
$\tfrac{1}{2} \cdot 8 \cdot 7.5 = 30 \text{ cm}^2, \qquad 30 \cdot 14 = 420 \text{ cm}^3$

Answer

The height of each triangular base is $7.5$ cm.

A word problem wraps the same relationship in a situation. Sometimes it asks for the volume. Sometimes it gives the volume and asks for a dimension. Either way, sort out four things before you calculate:

Base shape. Which face keeps the same shape and size all the way through the solid?
Base dimensions. Which lengths give the area of that face?
Prism height. How far apart are the two bases, measured at a right angle to them? A situation may call this distance a depth or a length.
Units. Every length must use the same unit, so that areas come out in that unit squared and volumes in that unit cubed.

Then write $V = B \cdot h$ , fill in what you know, and solve for what is missing. Finish by checking that the size and the unit of your answer fit the situation.

Finding the base and the prism height in a situation

Situation	Base	Prism height
Water in an upright tank with a level rectangular floor	The rectangular floor	The depth of the water, at a right angle to the floor
Soil spread level inside an upright rectangular frame on flat ground	The rectangle of ground inside the frame	The depth of the soil, at a right angle to the ground
A trough or tent shaped like a right triangular prism	One triangular end	The length from one triangular end to the other, at a right angle to both ends

Worked example

Pouring water from a trough into a tank

A water trough is a right triangular prism with a triangle at each end. Each triangular end has a top edge $40$ cm long and a height of $30$ cm, measured at a right angle from that edge down to the bottom corner. The trough is $150$ cm long, measured at a right angle to both ends, and it is full of water. You pour all the water into an empty upright tank with a level rectangular floor $100$ cm by $60$ cm. The tank is tall enough to hold it. How deep is the water in the tank?

Sort out the trough. Its bases are the two triangular ends: each has a 40 cm side and a perpendicular triangle height of 30 cm. The prism height is the 150 cm length, because it is the perpendicular distance between the two ends. Every length is in cm. Find the area of one end: one half of the side times the triangle height.
$\text{area of one end} = \tfrac{1}{2} \cdot 40 \cdot 30 = 20 \cdot 30 = 600 \text{ cm}^2$
A right prism's volume is its base area times its prism height, so multiply 600 by 150. Splitting 150 into 100 and 50 makes the product easier to find.
$\text{volume} = 600 \cdot 150 = 60{,}000 + 30{,}000 = 90{,}000 \text{ cm}^3$
Sort out the tank. Pouring all the water changes its shape but not its amount, so the volume is still 90,000 cubic centimeters. In the tank the water forms a right rectangular prism whose base is the floor. Find the floor's area: length times width.
$\text{area of floor} = 100 \cdot 60 = 6000 \text{ cm}^2$
Let d be the depth of the water in cm, measured at a right angle to the floor. That depth is the height of the water's right prism, so volume equals floor area times d.
$90{,}000 = 6000 \cdot d$
Divide both sides by 6000, which is not zero, so that d stands alone. Dividing the top and the bottom of the fraction by 1000 gives 90 divided by 6, which is 15. Cubic centimeters divided by square centimeters give centimeters.
$d = \frac{90{,}000}{6000} = \frac{90}{6} = 15 \text{ cm}$
Check by multiplying back. The size makes sense too: the floor has 10 times the area of a trough end, so the same volume needs only one tenth of the trough's 150 cm prism height.
$6000 \cdot 15 = 90{,}000 \text{ cm}^3$

Answer

The water in the tank is $15$ cm deep.

Takeaway. For a right prism with positive measures in one consistent unit, volume is base area times perpendicular height: $V = B \cdot h$ . When the volume is known, divide by the part you know to find the part you are missing: $h = V \div B$ or $B = V \div h$ . You never divide by zero here, because areas and heights are positive. If the missing dimension is a side of the base, find $B$ first and then use the area rule for the base shape. In a word problem, name the base shape, its dimensions, the prism height and the units before you calculate, and check by multiplying back.

Next, you will solve problems that use both surface area and volume, and you will practice finding and fixing errors in volume work.