Volume: Missing Dimensions, Word Problems, Error Checks | Grade 8 Math | LevelKite

You already know how to start with the measurements of a cylinder or a cone and find its volume. Real questions often run the other way. A candle maker knows how much wax she has and how wide her mold is, and she wants to know how deep the wax will be.

In this lesson you start with the volume and find a missing dimension, which is a measurement such as a radius or a height. You will also solve word problems, and you will catch and fix mistakes in volume work.

Here is what each letter means for a cylinder or a cone. Both are solids, which are shapes that take up space.

$r$ is the radius of the base, the flat circle at one end of the solid. The radius is the distance from the center of that circle to its edge. The diameter $d$ is the distance across the circle through its center, so $d = 2r$ . Both are positive lengths.
$h$ is the height: the distance from the base to the far end of the solid (a cylinder's other base or a cone's tip), measured at a right angle to the base. A length at a right angle to the base is perpendicular to it. The height is a positive length.
$V$ is the volume: the amount of space inside the solid.

Measure $r$ and $h$ in the same unit of length. Then the area of the base is in square units and $V$ is in cubic units. With centimeters, for example, $V$ is in cubic centimeters.

With those meanings, these are the two formulas you already know:

\text{Cylinder: } V = \pi r^2 h \qquad \text{Cone: } V = \tfrac{1}{3}\pi r^2 h

Read the cylinder formula as a sentence. Because $\pi r^2$ is the area of the circular base, $V = \pi r^2 h$ says volume equals base area times height, where the height is perpendicular to the base.

Now suppose you know the volume but not the height. Say a cylinder's base has an area of $12$ square centimeters and its volume is $84$ cubic centimeters. Finding the height means answering this: $12$ times what number gives $84$ ? Division answers that question. Since $84 \div 12 = 7$ , the height is $7$ centimeters.

A cone's volume is one third of base area times height. So $3$ times the cone's volume equals base area times height, and you can ask the same question from there.

Finding a radius takes one more tool, because the formula squares $r$ . The square root of $36$ , written $\sqrt{36}$ , is the positive number whose square is $36$ , so $\sqrt{36} = 6$ . A radius is a positive length, so taking the square root of $r^2$ gives back $r$ .

Key concept

A volume formula works in both directions

A volume formula is an equation that links three quantities: $V$ , $r$ and $h$ . When you know two of them, you can find the third. To find a missing radius or height, substitute the values you know, which means replace each known letter with its number. Then undo the operations on the unknown, one at a time, doing the same thing to both sides.

This works when $r$ and $h$ are positive lengths in the same unit, $h$ is perpendicular to the base (never a slant height), any diameter has been changed to a radius using $d = 2r$ , and $V$ is in the matching cubic unit.

How to undo each operation on an unknown positive length

What was done to the unknown	How to undo it	Why the equation stays true
Multiplied by a positive number, such as $16\pi$	Divide both sides by that number	The number is not zero, and both sides are divided by the same amount
Multiplied by $\frac{1}{3}$ (cone formula)	Multiply both sides by $3$	$3 \cdot \frac{1}{3} = 1$ , and both sides are multiplied by the same amount
Squared, as in $r^2$	Take the positive square root of both sides	Equal positive numbers have equal square roots, and $\sqrt{r^2} = r$ because $r$ is positive

Worked example

Missing height of a cylinder

A cylinder has a volume of $80\pi$ cubic centimeters and a radius of $4$ centimeters. What is its height $h$ ?

Write the cylinder formula and substitute the two values you know, the volume and the radius. The height h is the only unknown left.
$80\pi = \pi (4)^2 h$
Square only the radius, because the small 2 is attached to the radius and not to pi or h. Since 4 times 4 is 16, the equation now says that the base area times the height equals the volume.
$80\pi = 16\pi h$
The height is multiplied by the base area, so divide both sides by the base area. It is not zero, and both sides are divided by the same amount, so the equation stays true. Pi divided by pi is 1, so pi drops out.
$h = \frac{80\pi}{16\pi} = \frac{80}{16} = 5$
Check by putting 5 back in for the height. The result matches the given volume. The answer is a length, so its unit is centimeters: cubic centimeters divided by square centimeters leaves centimeters.
$\pi (4)^2 (5) = \pi \cdot 16 \cdot 5 = 80\pi$

Answer

The height of the cylinder is $5$ centimeters.

Worked example

Missing radius of a cone

A cone has a volume of $96\pi$ cubic inches and a height of $8$ inches, measured perpendicular to its base. What is its radius $r$ ?

Write the cone formula and substitute the volume and the height. The radius r is the unknown.
$96\pi = \tfrac{1}{3}\pi r^2 (8)$
Multiply both sides by 3 to undo the one third. Both sides are multiplied by the same number, so they stay equal. On the left, 3 times 96 is 288. On the right, 3 times one third is 1, so what remains is 8 times pi times r squared.
$288\pi = 8\pi r^2$
The unknown part, r squared, is multiplied by a number that is not zero. Divide both sides by that number. Pi drops out, and 288 divided by 8 is 36.
$r^2 = \frac{288\pi}{8\pi} = \frac{288}{8} = 36$
Take the positive square root of both sides. Both 6 and negative 6 have a square of 36, but a radius is a positive length, so only 6 fits.
$r = \sqrt{36} = 6$
Check by putting 6 back in for the radius. The result matches the given volume. The radius is a length, so its unit is inches.
$\tfrac{1}{3}\pi (6)^2 (8) = \tfrac{1}{3}\pi (36)(8) = \tfrac{1}{3}\pi (288) = 96\pi$

Answer

The radius of the cone is $6$ inches.

A word problem hides the same equation inside a story. Before you calculate, answer four questions.

Which solid is it? Decide whether the object is shaped like a cylinder or like a cone. That tells you which formula to use.
What does the question ask for? A phrase such as how much it holds usually asks for a volume. A phrase such as how tall or how deep usually asks for a missing dimension.
Which numbers are $r$ and $h$ ? A distance all the way across a circle, through its center, is a diameter $d$ . Since $d = 2r$ , halve a given diameter to get $r$ , and double $r$ if the question asks for a diameter. The height must be perpendicular to the base, so leave out a slant height.
What are the units? Use one unit of length for $r$ and $h$ . A volume gets cubic units. A missing radius or height gets a unit of length, not a cubic unit.

An answer that keeps the symbol $\pi$ is exact. If you replace $\pi$ with the rounded value $3.14$ , the result is an approximation, a value that is close but not exact. From that step on, write $\approx$ instead of $=$ .

Worked example

A word problem with a missing height

Now answer the candle maker's question. She pours $942$ cubic centimeters of melted wax into a tall mold shaped like a cylinder. The mold is $10$ centimeters across. About how deep is the wax? Use $3.14$ for $\pi$ .

Plan first. The wax takes the shape of the mold, so it forms a cylinder. Its depth is that cylinder's height h, measured straight up from the flat bottom, perpendicular to it. The question asks how deep, so the height is the missing dimension.
$V = \pi r^2 h$
The 10 centimeters is the distance across the circle, so it is the diameter. The formula needs the radius, which is half the diameter.
$r = 10 \div 2 = 5$
Substitute the volume, the radius and 3.14 for pi. The values are approximate from here on, so use the approximately-equal sign. Square the radius, then multiply by 3.14 to get the base area.
$942 \approx 3.14 \cdot 5^2 \cdot h = 3.14 \cdot 25 \cdot h = 78.5h$
Divide both sides by 78.5, which is not zero, to undo the multiplication.
$h \approx \frac{942}{78.5} = 12$
Check by multiplying back: the base area times 12 returns the 942 you started with. The depth is a length, so its unit is centimeters.
$78.5 \cdot 12 = 942$

Answer

The wax is about $12$ centimeters deep in the mold.

You can use the same thinking to check finished work, whether it is yours or someone else's. A wrong answer often comes from one small choice near the start, so read the work line by line and ask the questions in the table below. Keep checking after you find an error, because one calculation can contain more than one.

Questions that catch common errors in volume work

Question to ask	Error it catches	Example of the error
Does the formula match the solid?	Leaving out $\frac{1}{3}$ for a cone, or using it for a cylinder	Cone with $r = 2$ and $h = 9$ : writing $\pi(2)^2(9)$
Is the squared number the radius?	Using a diameter $d$ where the radius $r = d \div 2$ belongs	Circle $10$ centimeters across: writing $10^2$ instead of $5^2$
Is $h$ perpendicular to the base?	Using a cone's slant height as its height	Height $12$ meters, slant height $13$ meters: using $13$ for $h$
Was the radius squared?	Doubling the radius instead of multiplying it by itself	Writing $7^2 = 14$ instead of $7^2 = 49$
Do the units fit?	Mixing units of length, or putting the wrong kind of unit on the answer	Labeling a volume in centimeters instead of cubic centimeters
Does a missing dimension give back the volume?	A slip while undoing, such as dividing by $3$ instead of multiplying by $3$	A cone radius of $2$ that gives $\frac{32\pi}{3}$ , not the stated $96\pi$

Worked example

Find and fix the errors

A play tent is shaped like a cone. It is $6$ feet across its circular floor and $4$ feet tall, measured straight up from the floor to the tip. Each pole along its slanted side is $5$ feet long. Tia finds the space inside the tent: $V = \frac{1}{3}\pi(6)^2(5) = \frac{1}{3}\pi(36)(5) = 60\pi$ cubic feet. Find and correct her errors.

Check the formula. The tent is a cone, and Tia's first line has the one third, so the formula matches the solid.
$V = \tfrac{1}{3}\pi r^2 h$
Check the radius. The 6 feet is the distance across the floor, so it is the diameter. Tia squared the diameter. The radius is half of it. This is her first error.
$r = 6 \div 2 = 3$
Check the height. Tia used 5 feet, but a pole runs along the slanted side, so 5 feet is the slant height. The height perpendicular to the floor is 4 feet. This is her second error.
$h = 4$
Redo the work with the radius and the perpendicular height. Square the radius first, then multiply by the height, then take one third.
$V = \tfrac{1}{3}\pi (3)^2 (4) = \tfrac{1}{3}\pi (9)(4) = \tfrac{1}{3}\pi (36) = 12\pi$
Check the unit. Feet times feet times feet gives cubic feet, so Tia's unit was right. Her arithmetic was right too. Only her two starting values were wrong, and they made her answer 5 times as large as the correct one.
$\frac{60\pi}{12\pi} = 5$

Answer

Tia used the diameter as the radius and the slant height as the height. The correct volume is exactly $12\pi$ cubic feet.

Takeaway. A volume formula is one equation that links $V$ , $r$ and $h$ . Use it forward to find a volume. Or substitute what you know and undo the operations, the same way on both sides, to find a missing radius or height.

Before you trust an answer, check that the formula matches the solid, that $r$ is the radius (half the diameter, since $d = 2r$ ), that $h$ is perpendicular to the base and not a slant height, that all lengths share one unit so the volume is in cubic units, and that a missing dimension gives back the stated volume.

Next, you will use both formulas to compare the volume of a cylinder with the volume of a cone.