Volume of Cylinders, Cones, and Spheres

What Is Volume? 🧊

Volume measures the amount of space inside a 3D shape. You can think of it as:

• How much liquid the shape can hold
• How many unit cubes fit inside
• The capacity of the shape

Because volume fills up 3D space, we always measure it in cubic units: $\text{cm}^3$, $\text{in}^3$, $\text{m}^3$, or $\text{ft}^3$.

Curved 3D shapes are everywhere in real life:

• A soda can is a cylinder 🥤
• An ice cream cone is a cone 🍦
• A basketball is a sphere 🏀

Once you learn three formulas, you can find the volume of all of them. The secret is that they're all connected through the circle area $\pi r^2$.

Volume of a Cylinder 🥫

A cylinder has two circular bases and a curved side, like a soda can.

Formula:

$V = \pi r^2 h$

Where:

• $r$ = radius of the circular base
• $h$ = height of the cylinder
• $\pi \approx 3.14$ (or use your calculator's $\pi$ button)

Why does it work? Volume of a cylinder is just the area of the base times the height:

• Base area (a circle) $= \pi r^2$
• Multiply by height: $\pi r^2 \times h = \pi r^2 h$

Worked Example. Find the volume of a cylinder with radius $3$ cm and height $10$ cm. Use $\pi \approx 3.14$.

$V = \pi r^2 h = \pi \times (3)^2 \times 10 = \pi \times 9 \times 10 = 90\pi$

$V \approx 90 \times 3.14 \approx 282.6 \text{ cm}^3$

Watch out! Square the radius before multiplying by the height. The exponent only touches the $r$.

Cones and Spheres 🍦🏀

Two more shapes complete the set. Notice how each formula is built from $\pi r^2$ or $r^3$.

Shape	Formula	Key Idea
Cylinder	$V = \pi r^2 h$	base area $\times$ height
Cone	$V = \frac{1}{3}\pi r^2 h$	exactly $\frac{1}{3}$ of a matching cylinder
Sphere	$V = \frac{4}{3}\pi r^3$	depends only on the radius

Cone insight: A cone holds exactly one-third of a cylinder with the same base and same height. That's where the $\frac{1}{3}$ comes from!

Sphere insight: A sphere has no height to plug in — its size depends only on the radius. Notice the radius is cubed ($r^3$, not $r^2$), because a sphere curves in every direction.

Quick cone example. Cone with $r = 5$ in and $h = 9$ in: $V = \frac{1}{3}\pi (5)^2 (9) = \frac{1}{3}\pi (25)(9) = \frac{1}{3}\pi (225) = 75\pi \approx 235.5 \text{ in}^3$

Concept Check ✅

Let's make sure the big ideas are clear before moving on.

Worked Example: A Water Tank 💧

A cylindrical water tank has diameter $8$ ft and height $12$ ft. What is its volume? Use $\pi \approx 3.14$.

Step 1 — Find the radius. The problem gives the diameter, but the formula needs the radius. The radius is half the diameter: $r = \frac{8}{2} = 4 \text{ ft}$

Step 2 — Plug into the formula. $V = \pi r^2 h = \pi (4)^2 (12)$

Step 3 — Simplify in order. Square first, then multiply: $V = \pi (16)(12) = 192\pi$

Step 4 — Estimate with $\pi \approx 3.14$. $V \approx 192 \times 3.14 \approx 603.2 \text{ ft}^3$

Answer: $192\pi \text{ ft}^3$, or about $603 \text{ ft}^3$. The most common mistake is forgetting to cut the diameter in half — don't fall for it!

Worked Example: A Sphere 🌐

Find the volume of a sphere with radius $3$ cm. Use $\pi \approx 3.14$.

Step 1 — Write the formula. $V = \frac{4}{3}\pi r^3$

Step 2 — Cube the radius. Remember, $r^3 = r \times r \times r$: $3^3 = 3 \times 3 \times 3 = 27$

Step 3 — Multiply. $V = \frac{4}{3}\pi (27) = \frac{4 \times 27}{3}\pi = \frac{108}{3}\pi = 36\pi$

Step 4 — Estimate. $V \approx 36 \times 3.14 \approx 113.04 \text{ cm}^3$

Answer: $36\pi \text{ cm}^3$, or about $113 \text{ cm}^3$. A friendly trick: dividing $27$ by $3$ first ($= 9$) keeps the numbers small: $4 \times 9 = 36$.

Your Turn — Fill in the Boxes ✍️

Find the volume of a cone with radius $2$ cm and height $12$ cm. Use $\pi \approx 3.14$.

Leave Boxes 1 and 2 as exact numbers (no $\pi$). For Box 3, write the volume as the number in front of $\pi$ (the "$\pi$-coefficient").

Box 1: The value of $r^2$ (that is, $2^2$) Box 2: The value of $r^2 \times h$ (multiply Box 1 by the height $12$) Box 3: The volume as a coefficient of $\pi$ — take $\frac{1}{3}$ of Box 2

Guided Practice: Choose the Answer 🎯

Pick the right formula, plug in carefully, and simplify. Answers are left in terms of $\pi$.

Match the Shape to Its Formula 🔽

Choose the correct volume formula for each shape.

Volume in the Real World 🌍

These formulas help people make real decisions about capacity — how much something holds.

Example — a grain silo. A farmer's cylindrical silo has radius $5$ m and height $14$ m. How much grain can it hold? Use $\pi \approx 3.14$.

• $V = \pi r^2 h = \pi (5)^2 (14) = \pi (25)(14) = 350\pi$
• $V \approx 350 \times 3.14 \approx 1099 \text{ m}^3$

So the silo holds about $1{,}099 \text{ m}^3$ of grain.

Reading word problems carefully matters:

• If a problem gives the diameter, cut it in half to get the radius.
• A "ball," "globe," or "marble" is a sphere.
• A "can," "tank," "pipe," or "drum" is a cylinder.
• A "funnel," "party hat," or "scoop holder" is a cone.

Now try a couple of real-world problems yourself!

Word Problem — Fill in the Boxes ✍️

A spherical bouncy ball has diameter $6$ cm. Find its volume in terms of $\pi$.

Box 1: The radius of the ball in cm (half the diameter) Box 2: The value of $r^3$ (cube your Box 1 answer) Box 3: The volume as a coefficient of $\pi$ — take $\frac{4}{3}$ of Box 2

Application Question 🧮

Read carefully and choose the correct shape and formula.

Summary: All Three Formulas 🎓

You can now find the volume of any cylinder, cone, or sphere. Here's everything in one place:

Shape	Formula	What you need	Remember
Cylinder 🥫	$V = \pi r^2 h$	radius and height	base area $\times$ height
Cone 🍦	$V = \frac{1}{3}\pi r^2 h$	radius and height	$\frac{1}{3}$ of a cylinder
Sphere 🏀	$V = \frac{4}{3}\pi r^3$	radius only	radius is cubed

Three habits that prevent mistakes:

• Diameter → radius. If you're given the diameter, divide by $2$ first.
• Exponent order. Square (or cube) the radius before multiplying by anything else.
• Right units. Volume always ends in cubic units like $\text{cm}^3$ or $\text{ft}^3$.

The two final questions mix all three shapes — read carefully and pick the right formula each time!

Final Challenge 🏆

Mix and match everything you've learned. Answers are in terms of $\pi$.