8th Grade · 8.G.C.9

Volume of cones, cylinders and spheres in 8th grade

In 8th grade geometry, children learn to find the volume of cones, cylinders and spheres. Volume is the amount of space inside a solid shape, measured in cubic units such as cubic inches. Your child learns one formula for each shape and uses it to solve problems about cans, cones and balls.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

8.G.C.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

What it means, in plain words

All three shapes are round, so all three formulas use $\pi$ (pi), a number close to 3.14 that comes up in every circle. They also use the radius, $r$ , which is the distance from the center of a circle to its edge. The diameter is the full width across, twice the radius.

The volume of a cylinder is the area of its circle base times its height: $V = \pi r^2 h$ . A cone with the same base and height holds exactly one third of that cylinder, so its formula is $V = \frac{1}{3}\pi r^2 h$ . The volume of a sphere (a ball shape) is $V = \frac{4}{3}\pi r^3$ , where $r^3$ means $r \times r \times r$ .

Every problem about the volume of cones, cylinders and spheres comes back to these three formulas. Teachers often ask for the answer in terms of $\pi$ , such as $36\pi$ cubic inches, which keeps $\pi$ as a symbol. Other problems want a decimal, using 3.14 for $\pi$ . Your child also works backward: given the volume, find the height or the radius.

Your child should be able to:

Find the volume of a cylinder, a cone and a sphere from a picture or a word problem
Tell the radius from the diameter and put the radius in the formula
Give the answer in terms of $\pi$ or as a rounded decimal, in cubic units
Work backward from a volume to find a missing height or radius

Example

A can is a cylinder with a radius of 3 cm and a height of 10 cm. What is its volume, and what is the volume of a cone with the same radius and height?

Cylinder: $V = \pi \times 3^2 \times 10 = \pi \times 9 \times 10 = 90\pi$
Cone: one third of the cylinder, so $V = \frac{1}{3} \times 90\pi = 30\pi$
Using 3.14 for $\pi$ , the can holds about 282.6 cubic centimeters and the cone holds about 94.2 cubic centimeters.

Example

A ball has a radius of 3 inches. What is its volume?

$r^3 = 3 \times 3 \times 3 = 27$
$V = \frac{4}{3} \times \pi \times 27 = 36\pi$
Using 3.14 for $\pi$ , $36 \times 3.14 = 113.04$ , so the ball holds about 113 cubic inches.

Where children usually get stuck

A cylinder is labeled 6 inches across, and your child puts 6 into the formula as the radius.
Pictures often show the diameter, the full width. The formula needs the radius, half of that width. Using the diameter makes the volume of a cylinder or a cone four times too big.
What helps. Before any numbers, ask your child to point to the radius on the picture. If the label shows the full width, cut it in half first.
Your child gets the same volume for a cone and a cylinder with the same radius and height.
The cone formula is the cylinder formula with $\frac{1}{3}$ in front, and that small fraction is easy to leave out.
What helps. Hold up an ice cream cone next to a cup of about the same width and height, and ask which holds more. The cone holds about a third as much, which is where the $\frac{1}{3}$ comes from.
For a sphere with a radius of 3, your child writes $r^3 = 9$ .
Squaring is more familiar than cubing, so $3^3$ gets worked out as $3 \times 3$ . A cube needs three factors: $3 \times 3 \times 3 = 27$ .
What helps. Ask your child to write $r^3$ out as three numbers multiplied together every time, before working it out. Seeing three 3s on the page stops the mix-up.

Try four practice questions

Real 8th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
A cylinder and its volume are shown below. What is the height?
$\displaystyle V = 12\pi$
A
3
B
2
C
4
D
6
Show the answer and how to solve it
The height is 3. The base area is $\pi r^{2} = 4\pi$ , so $h = \frac{12\pi}{4\pi} = 3$ .
1. 1The cylinder shows a radius of 2, and the volume is $12\pi$ .
2. 2Use the cylinder volume formula.
3. 3Substitute $12\pi$ for $V$ and 2 for $r$ . Divide both sides by $\pi$ , then by 4.
Common mistake. After canceling $\pi$ , divide 12 by $r^{2} = 4$ , not by $r = 2$ . Using $12 \div 2 = 6$ gives the wrong height.
The idea underneath. When the height is missing, dividing a cylinder's volume by $\pi$ and $r^{2}$ isolates $h$ directly — no square root needed.
2On level
The cone below has a volume of $27\pi$ cubic units. What is its height?
Height:units
Show the answer and how to solve it
Answer. 9
The answer is $9$ units because $27\pi = \frac{1}{3}\pi(3^{2})h$ , and solving that equation gives $h = 9$ .
1. 1Look at the cone. The diagram shows a radius of $3$ units, and the problem says the volume is $27\pi$ cubic units.
2. 2Use the volume formula for a cone.
3. 3Substitute $V = 27\pi$ and $r = 3$ .
4. 4Solve for $h$ .
5. 5So the height of the cone is $9$ units.
Common mistake. A common mistake is using the cylinder formula $V = \pi r^{2}h$ instead of the cone formula. That would give $h = 3$ , but a cone has only one-third the volume of a cylinder with the same base and height.
The idea underneath. A cone with the same base and height as a cylinder has one-third the volume, so solving for height means undoing both the $\frac{1}{3}$ factor and the squared radius.

3On level

Owen used the volume of a cylinder to find the radius. Which statement correctly identifies the mistake in the work below?

Formula	$\displaystyle V$	$\displaystyle =$	$\displaystyle \pi r^2 h$
Substitute	$\displaystyle 80\pi$	$\displaystyle =$	$\displaystyle \pi r^2(5)$
Divide by π	$\displaystyle 80$	$\displaystyle =$	$\displaystyle 5r^2$
Divide by 5	$\displaystyle 16$	$\displaystyle =$	$\displaystyle r^2$
Owen's answer	$\displaystyle r$	$\displaystyle =$	$\displaystyle 16$

Owen used r instead of

r^{2}

. The radius should stay squared in the cylinder volume formula.

Owen canceled π incorrectly. Dividing both sides by π was not valid.

Owen found

r^{2} = 16

, but forgot to take the square root to solve for r.

Owen squared the wrong dimension. The height should have been squared instead of the radius.

Show the answer and how to solve it

The correct statement is that Owen forgot to take the square root after finding $r^{2} = 16$ . The radius is 4, not 16.

1Owen used the cylinder volume formula $V = \pi r^{2}h$ and substituted $V = 80\pi$ and $h = 5$ . The setup is correct.
2Dividing both sides by $\pi$ and then by 5 is valid, so the work correctly reaches $r^{2} = 16$ .
3To solve for $r$ , you must take the square root of both sides. $\sqrt{16} = 4$ , so the radius is 4, not 16.
4The mistake is in the last line: Owen treated $r^{2} = 16$ as if it already gave $r$ , writing r = 16 instead of completing the square root step.

Common mistake. A common mistake is stopping at $r^{2} = 16$ and thinking the radius is 16. Always check whether you solved for $r$ or only for $r^{2}$ .

The idea underneath. When solving a formula, you are not finished until the missing measure itself is isolated; getting $r^{2}$ means one more inverse operation is still needed.

4Stretch
Drag the solids so they are ordered from least to greatest volume.
Drag a card to move it, or tap a card, then tap where it should go.
1
2
3
4
Show the answer and how to solve it
From least to greatest: sphere $r=2$ ( $\frac{32\pi}{3}$ ), cone $r=3, h=4$ ( $12\pi$ ), cone $r=4, h=9$ ( $48\pi$ ), cylinder $d=6, h=6$ ( $54\pi$ ).
1. 1Find the volume of each solid. The cylinder shows a diameter label, so convert to radius first. The sphere produces a fraction of $\pi$ , so multiply every coefficient by 3 to compare integers.
2. 2For the cylinder ( $d = 6$ , $h = 6$ ), convert: $r = \frac{6}{2} = 3$ . Then compute its volume.
3. 3For the cone with $r = 3$ and $h = 4$ :
4. 4For the cone with $r = 4$ and $h = 9$ :
5. 5For the sphere with $r = 2$ :
6. 6Multiply each coefficient by 3 and compare: $32 < 36 < 144 < 162$ .
Common mistake. A common mistake is using the diameter value 6 directly as the radius for the cylinder, yielding $\pi(6^2)(6) = 216\pi$ instead of $54\pi$ . Always halve the diameter before substituting into the formula.
The idea underneath. Because every volume has the same positive factor $\pi$ , you can compare the simplified coefficients instead of estimating with a decimal for $\pi$ .