Volume of a Ball

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.

Starting From the Diameter

A globe has a diameter of $12$ cm. Find its volume in terms of $\pi$, then use $\pi \approx 3.14$ and round to the nearest hundredth.

Find the radius (half the diameter): $r = 12 \div 2 = 6$ = $r = 6$ cm

Cube the radius: $6^3 = 6 \times 6 \times 6 = 216$ = $r^3 = 216$

Multiply by $\frac{4}{3}$: $216 \div 3 = 72$, then $72 \times 4 = 288$ = $V = 288\pi$ cm³

Replace $\pi$ with $3.14$: $288 \times 3.14 = 904.32$ = $V \approx 904.32$ cm³

Answer: The globe has a volume of $288\pi$ cm³, about $904.32$ cm³. Using $12$ as the radius would give a volume $8$ times too big!

Volume of a Hemisphere

A bowl is shaped like a hemisphere with an inside radius of $6$ cm. How much soup can it hold? Use $\pi \approx 3.14$ and round to the nearest hundredth.

Write the hemisphere formula (half of a sphere): $V = \frac{2}{3}\pi r^3$ = Formula ready

Cube the radius: $6^3 = 216$ = $r^3 = 216$

Multiply by $\frac{2}{3}$: $216 \div 3 = 72$, then $72 \times 2 = 144$ = $V = 144\pi$ cm³

Replace $\pi$ with $3.14$ and change to mL ($1$ cm³ $= 1$ mL): $144 \times 3.14 = 452.16$ = $V \approx 452.16$ cm³ $\approx 452$ mL

Answer: The bowl holds $144\pi$ cm³, about $452.16$ cm³, which is roughly $452$ mL of soup. Check: it is exactly half of the globe in Example 2 ($288\pi \div 2 = 144\pi$).

Finding the Radius From the Volume

A sphere has a volume of $4500\pi$ cm³. What is its radius?

Set the formula equal to the volume: $\frac{4}{3}\pi r^3 = 4500\pi$ = Equation ready

Divide both sides by $\pi$: $\frac{4}{3} r^3 = 4500$ = $\pi$ is gone

Multiply both sides by $\frac{3}{4}$: $r^3 = 4500 \times \frac{3}{4} = 3375$ = $r^3 = 3375$

Take the cube root (which number times itself three times is $3375$?): $15 \times 15 \times 15 = 3375$ = $r = 15$ cm

Answer: The radius is $15$ cm. Check: $\frac{4}{3} \times 15^3 = \frac{4}{3} \times 3375 = 4500$, so $V = 4500\pi$ cm³.

Archimedes: Sphere Inside a Cylinder

A ball with radius $3$ cm fits exactly inside a can: the can has radius $3$ cm and height $6$ cm. What fraction of the can does the ball fill?

Volume of the can (cylinder): $V = \pi r^2 h$: $\pi \times 3^2 \times 6 = \pi \times 9 \times 6$ = $V_{\text{can}} = 54\pi$ cm³

Volume of the ball: $\frac{4}{3} \times 27 = 36$ = $V_{\text{ball}} = 36\pi$ cm³

Compare the two volumes: $\frac{36\pi}{54\pi} = \frac{36}{54} = \frac{2}{3}$ = The ball fills $\frac{2}{3}$ of the can

Answer: The ball fills $\frac{2}{3}$ of the can and $\frac{1}{3}$ (here $18\pi$ cm³) stays empty. This works for every sphere in its matching cylinder: Archimedes' discovery.

Mistake: Using the diameter instead of the radius

Why: 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 $r$.

Mistake: Squaring the radius instead of cubing it

Why: 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.

Mistake: Halving the radius for a hemisphere instead of halving the volume

Why: 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 $\frac{4}{3}\pi r^3$).

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

Why: 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 $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000$.

Sports Balls

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

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.

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.

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).

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.

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.

Q: Where does the $\frac{4}{3}$ come from?

A: 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: $\frac{2}{3} \times 2\pi r^3 = \frac{4}{3}\pi r^3$.

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

A: 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.

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

A: 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 $r = 6$.

Q: If I double the radius, does the volume double?

A: 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.