🎯⭐ INTERACTIVE LESSON

Volume of Cylinders, Cones, and Spheres

Learn step-by-step with interactive practice!

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Volume of Cylinders, Cones, and Spheres - Complete Interactive Lesson

Part 1: What Is Volume? 🧊

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
(or use your calculator's button)

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	$V = π r^{2} h$

Concept Check ✅

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

Part 2: Worked Example: A Water Tank 💧

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: $r = \frac{8}{2} = 4 \text{ ft}$

Part 3: Guided Practice: Choose the Answer

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.

Part 4: Volume in the Real World 🌍

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 $\pi \approx 3.14$ .

$V = π r^{2} h = π (5)^{2} (14) = π (25) ($

Part 5: Summary: All Three Formulas 🎓

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

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 $r$ .

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: $V = \frac{1}{3}\pi (5)^2 (9) = \frac{1}{3}\pi (25)(9) = \frac{1}{3}\pi (225) = 75\pi \approx 235.5 \text{ in}^3$

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 () keeps the numbers small: .

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 ) The volume as a coefficient of — take of Box 2

V = π r^{2} h = π (5)^{2} (14) = π (25) (14) = 350 π

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.

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 $\pi$ .