Should I leave the answer in terms of $\pi$ or use $3.14$?

Do what the problem asks. An answer like $36\pi$ cm³ is exact. With $3.14$ you get an approximate decimal ($113.04$ cm³). With the $\pi$ key on a calculator the result is slightly different ($113.10$ cm³), so always say which value of $\pi$ you used and how you rounded.

How do I find the radius if I know the volume?

Undo the formula step by step: divide the volume by $\pi$, multiply by $\frac{3}{4}$ to get $r^3$, then find the cube root. For $V = 288\pi$: $288 \times \frac{3}{4} = 216$ and $\sqrt[3]{216} = 6$, so $r = 6$.

If I double the radius, does the volume double?

No, it becomes $8$ times bigger, because $2^3 = 8$. Tripling the radius makes the volume $27$ times bigger ($3^3 = 27$). That is why a beach ball holds so much more air than a tennis ball.

Teacher Resources

Volume of Spheres

Learn to find the volume of a sphere and a hemisphere with V = (4/3)πr³, and work backwards from a volume to the radius.

Intermediate25 minLesson

Printable worksheet

Definition

A sphere is a perfectly round 3D shape, like a ball. Every point on its surface is the same distance from the center. That distance is the radius

r

. The diameter

d

goes straight through the center, so

d = 2r

and

r = \frac{d}{2}

The volume of a sphere (the space inside it) is:

V = \frac{4}{3}\pi r^3

The radius is cubed (

r^3 = r \times r \times r

), because volume is measured in cubic units such as cm³ or m³.

A hemisphere is exactly half of a sphere, so its volume is half as much:

V_{\text{hemisphere}} = \frac{1}{2} \cdot \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3

You can give the answer in two ways:

In terms of π: keep the symbol, for example $V = 36\pi$ cm³ (this is exact).
As a decimal: use $\pi \approx 3.14$ and round as the problem asks, for example $36 \times 3.14 = 113.04$ cm³.

Try it now

Which formula gives the volume of a sphere with radius

r

Worked Examples

A ball has a radius of $3$ cm. Find its volume in terms of $\pi$ , then use $\pi \approx 3.14$ and round to the nearest hundredth.

Write the formula

$V = \frac{4}{3}\pi r^3$ → Formula ready

Cube the radius

$r^3 = 3 \times 3 \times 3 = 27$ → $r^3 = 27$

Multiply by $\frac{4}{3}$

$\frac{4}{3} \times 27 = \frac{108}{3} = 36$ → $V = 36\pi$ cm³

Replace $\pi$ with $3.14$

$36 \times 3.14 = 113.04$ → $V \approx 113.04$ cm³

Answer: The volume is $36\pi$ cm³, which is about $113.04$ cm³.

Common Mistakes

Using the diameter instead of the radius

Why it's wrong: The formula needs the radius. A sphere with diameter $10$ cm has radius $5$ cm. Using $10$ makes $r^3$ equal to $1000$ instead of $125$ , so the answer is $8$ times too big.

Correct: Always halve the diameter first: $r = \frac{d}{2}$ . Then cube $r$ .

Squaring the radius instead of cubing it

Why it's wrong: The area of a circle uses $r^2$ , so it is easy to write $r^2$ out of habit. But volume is three-dimensional and needs three factors of $r$ .

Correct: $V = \frac{4}{3}\pi r^3$ , so for $r = 3$ : $r^3 = 27$ , not $9$ . The unit is cm³, not cm².

Halving the radius for a hemisphere instead of halving the volume

Why it's wrong: A hemisphere has the same radius as the full sphere. Only the volume is cut in half. Halving the radius makes the volume $8$ times smaller, not $2$ times.

Correct: Hemisphere: keep $r$ and use $V = \frac{2}{3}\pi r^3$ (half of $\frac{4}{3}\pi r^3$ ).

Dividing by $3$ instead of taking the cube root when finding $r$

Why it's wrong: When you reach $r^3 = 216$ , the radius is not $216 \div 3 = 72$ . $r^3$ means $r \times r \times r$ , so you need the number that, multiplied by itself three times, gives $216$ .

Correct: $r = \sqrt[3]{216} = 6$ , because $6 \times 6 \times 6 = 216$ . Learn the cubes $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000$ .

Interactive Visual

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Radius (3)

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Edges

Vertices

Volume

V = ⁴⁄₃πr³

V = ⁴⁄₃π · 3³ = 36π ≈ 113.1

113.1 cm³

Surface Area

SA = 4πr²

SA = 4π · 3² = 36π ≈ 113.1

113.1 cm²

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Practice Problems

18 problems

Problem 1 of 18

Easy

Which formula gives the volume of a sphere with radius $r$ ?

Why It Matters

Round objects are everywhere, and their volume tells you how much they hold or how much material they need:

Sports: how much air is inside a football, a basketball or a tennis ball
Food: how much ice cream is in one scoop, or how many scoops come out of a tub
Science: soap bubbles, raindrops and water droplets are almost perfect spheres
Space: planets and stars are nearly spheres, so astronomers compare their sizes with this formula
Engineering: round gas tanks and domes (hemispheres) are built using these volumes

About 2,200 years ago, Archimedes discovered that a sphere fills exactly

\frac{2}{3}

of the cylinder that fits around it. He was so proud of it that he asked for a sphere and a cylinder to be carved on his tomb!

Real World Applications

Sports Balls

Ball makers use the sphere formula to know how much air a ball holds and how much material is needed inside it.

Example:

A tennis ball has a diameter of about $6.6$ cm, so $r = 3.3$ cm. Its volume is about $\frac{4}{3} \times 3.14 \times 3.3^3 \approx 150$ cm³.

1Try It Yourself

A handball has a diameter of $18$ cm.

What is its volume? Use $\pi \approx 3.14$ and round to the nearest hundredth.

Step 1: Write the mathematical expression

Halve the diameter, cube the radius, multiply by $\frac{4}{3}$ and by $3.14$ :

Planets and Space

Planets are almost perfect spheres. Because the radius is cubed, a planet with a bigger radius has a much, much bigger volume.

Example:

Earth's radius is about $6371$ km, so its volume is about $\frac{4}{3} \times 3.14 \times 6371^3 \approx 1.08 \times 10^{12}$ km³ (over a trillion cubic kilometers).

2Try It Yourself

The radius of Jupiter is about $11$ times the radius of Earth.

About how many Earths would fit into Jupiter by volume (if you could melt them and pour them in)?

Step 1: Write the mathematical expression

If the radius is multiplied by $11$ , the volume is multiplied by $11^3$ :

Ice Cream Scoops and Bubbles

An ice cream scoop makes balls of the same size, and a soap bubble is a thin skin around a sphere of air. Both are sphere volumes.

Example:

A soap bubble with radius $5$ cm holds about $\frac{4}{3} \times 3.14 \times 125 \approx 523.33$ cm³ of air, roughly half a liter.

3Try It Yourself

A tub holds $2$ liters ( $2000$ cm³) of ice cream. Each scoop is a sphere with radius $3$ cm.

How many full scoops can you serve? Use $\pi \approx 3.14$ .

Step 1: Write the mathematical expression

Divide the tub's volume by the volume of one scoop:

Key Takeaways

1The volume of a sphere is $V = \frac{4}{3}\pi r^3$ , where $r$ is the radius
2If you are given the diameter, halve it first: $r = \frac{d}{2}$
3A hemisphere is half a sphere: $V = \frac{2}{3}\pi r^3$ (same radius, half the volume)
4Give exact answers in terms of $\pi$ (like $36\pi$ cm³) or use $\pi \approx 3.14$ and round as asked
5To find $r$ from $V$ : divide by $\pi$ , multiply by $\frac{3}{4}$ , then take the cube root
6A sphere fills exactly $\frac{2}{3}$ of the cylinder that fits around it (Archimedes)

Frequently Asked Questions

From Archimedes' discovery. The cylinder that fits around a sphere has radius

r

and height

2r

, so its volume is

\pi r^2 \times 2r = 2\pi r^3

. The sphere fills

\frac{2}{3}

of it:

\frac{2}{3} \times 2\pi r^3 = \frac{4}{3}\pi r^3

Glossary

Sphere: A perfectly round 3D shape where every point on the surface is the same distance from the center
Radius ( $r$ ): The distance from the center of a sphere to its surface
Diameter ( $d$ ): The distance straight across a sphere through its center; $d = 2r$
Hemisphere: Exactly half of a sphere, cut through the center; its volume is $\frac{2}{3}\pi r^3$
Cube root: The number that, multiplied by itself three times, gives a value; $\sqrt[3]{27} = 3$
In terms of $\pi$: An exact answer that keeps the symbol $\pi$ , like $36\pi$ cm³

Formula Card

Volume of a sphere

V = \frac{4}{3}\pi r^3

Four thirds times pi times the radius cubed

Volume of a hemisphere

V = \frac{2}{3}\pi r^3

Half the volume of the sphere with the same radius

Radius from diameter

r = \frac{d}{2}

Halve the diameter before using the formula

Radius from volume

r = \sqrt[3]{\frac{3V}{4\pi}}

Divide by pi, multiply by 3/4, then take the cube root

Volume of Spheres

Definition

Worked Examples

Common Mistakes

Interactive Visual

3D Shape Viewer

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Sports Balls

Planets and Space

Ice Cream Scoops and Bubbles

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Volume of a sphere

Volume of a hemisphere

Radius from diameter

Radius from volume

More in This Topic

Introduction to Volume

Metric Volume Units (mL, L)

Volume of Rectangular Prisms

Imperial Volume Units (cups, pints, gallons)