Volume of Cylinders

Learn how to find the volume of a cylinder with V = πr²h, use the radius instead of the diameter, and convert cm³ to liters for cans, tanks and glasses.

Intermediate25 min18 problems
Practice Problems

Definition

A cylinder is a 3D shape with two identical, parallel circular bases joined by one curved surface. Think of a soda can, a candle or a water tank.
You already know that the volume of a prism is base area times height. A cylinder works the same way: stack thin circles on top of each other until you reach the height.
V=B×hV = B \times h
The base is a circle, and the area of a circle is πr2\pi r^2. So:
V=πr2hV = \pi r^2 h
Where:
  • rr = radius of the base (from the center to the edge)
  • hh = height (the distance between the two bases)
  • π≈3.14\pi \approx 3.14
Radius or diameter? The formula needs the radius. If you are given the diameter dd (all the way across), halve it first: r=d÷2r = d \div 2.
Two ways to give the answer:
  • **In terms of π\pi** (exact): keep π\pi as a symbol, for example 90π90\pi cm³.
  • As a number: use π≈3.14\pi \approx 3.14 and round to the place the problem asks for, for example 90×3.14=282.690 \times 3.14 = 282.6 cm³.
Liters: 1 cm3=1 mL1 \text{ cm}^3 = 1 \text{ mL}, so 1000 cm3=1 L1000 \text{ cm}^3 = 1 \text{ L}. For big tanks: 1 m3=1000 L1 \text{ m}^3 = 1000 \text{ L}.

Try it now

Which formula gives the volume of a cylinder with radius rr and height hh?

Worked Examples

A cylinder has radius 33 cm and height 1010 cm. Find its volume in terms of π\pi, then use π≈3.14\pi \approx 3.14.
1

Write the formula

V=πr2hV = \pi r^2 h → Formula ready
2

Square the radius

r2=32=9r^2 = 3^2 = 9 → r2=9r^2 = 9
3

Multiply by the height

V=π×9×10=90πV = \pi \times 9 \times 10 = 90\pi → V=90πV = 90\pi cm³ (exact)
4

Use π≈3.14\pi \approx 3.14

90×3.14=282.690 \times 3.14 = 282.6 → V≈282.6V \approx 282.6 cm³

Common Mistakes

Using the diameter instead of the radius

Why it's wrong: The formula squares the radius. If you square the diameter, the answer is 4 times too big. For d=10d = 10 cm and h=5h = 5 cm: π×102×5=500π\pi \times 10^2 \times 5 = 500\pi is wrong.

Correct: Halve the diameter first: r=5r = 5, so V=π×52×5=125πV = \pi \times 5^2 \times 5 = 125\pi cm³.

Doubling the radius instead of squaring it

Why it's wrong: r2r^2 means r×rr \times r, not r×2r \times 2. For r=3r = 3: 32=93^2 = 9, not 66.

Correct: Always multiply the radius by itself: r2=3×3=9r^2 = 3 \times 3 = 9.

Dividing by 100 instead of 1000 when converting to liters

Why it's wrong: A liter is a cube 1010 cm on each side, so it holds 10×10×10=100010 \times 10 \times 10 = 1000 cm³, not 100100.

Correct: 942000942000 cm³ ÷1000=942\div 1000 = 942 L. Remember: 11 cm³ =1= 1 mL.

Rounding too early

Why it's wrong: If you round the base area before multiplying by the height, the error grows. For r=2.5r = 2.5 cm and h=10h = 10 cm, rounding B=19.625B = 19.625 to 2020 gives 200200 cm³.

Correct: Keep all digits until the end: 19.625×10=196.2519.625 \times 10 = 196.25 cm³, then round only the final answer.

Interactive Visual

3D Shape Viewer

Faces

3

Edges

2

Vertices

0

Volume

V = πr²h

V = π · 3² · 10 = 90π ≈ 282.74

282.74 cm³

Surface Area

SA = 2πr² + 2πrh

SA = 2π · 3² + 2π · 3 · 10 = 78π ≈ 245.04

245.04 cm²

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which formula gives the volume of a cylinder with radius rr and height hh?

Why It Matters

Cylinders are everywhere, and their volume tells you how much they hold:
  • Drinks and food: how many milliliters fit in a can, a bottle or a glass?
  • Water tanks and rain barrels: how many liters can you store in the garden?
  • Pools: how much water do you need to fill a round paddling pool?
  • Engineering: engine cylinders, pipes and silos are all measured with V=πr2hV = \pi r^2 h.
  • Cooking and baking: a round cake pan holds a different amount than a square one.
Once you know one formula, you can answer all of these questions, and even work backwards to find how tall or how wide a container must be.

Real World Applications

Drink Cans

Drink companies design cans so they hold a set amount, like 330 mL. The volume formula tells them how tall and how wide the can must be.

Example:

A can with radius 3.33.3 cm and height 1010 cm holds about 3.14×3.32×10≈3423.14 \times 3.3^2 \times 10 \approx 342 cm³, enough for 330330 mL plus some air.

1Try It Yourself

A juice can is a cylinder with radius 33 cm and height 1212 cm.

How many milliliters does it hold? Use π≈3.14\pi \approx 3.14.

Step 1: Write the mathematical expression

Multiply 3.143.14 by the radius squared and by the height:

Rain Barrels

Gardeners collect rainwater in cylindrical barrels. Knowing the volume tells them how many liters they can save for dry weeks.

Example:

A barrel with radius 2525 cm and height 8080 cm holds 3.14×625×80=1570003.14 \times 625 \times 80 = 157000 cm³, which is 157157 L.

2Try It Yourself

A rain barrel has a diameter of 6060 cm and a height of 9090 cm.

How many liters of water can it hold? Use π≈3.14\pi \approx 3.14.

Step 1: Write the mathematical expression

Halve the diameter, then use V=πr2hV = \pi r^2 h and divide by 10001000:

Paddling Pools

Before filling a round paddling pool, it helps to know how much water you need. Here the height is the depth of the water, not the height of the pool wall.

Example:

A pool with radius 1.51.5 m filled to 0.20.2 m holds 3.14×2.25×0.2=1.4133.14 \times 2.25 \times 0.2 = 1.413 m³, which is 14131413 L.

3Try It Yourself

A round paddling pool has a diameter of 22 m. You fill it with water to a depth of 0.40.4 m.

How many liters of water do you need? Use π≈3.14\pi \approx 3.14 and 11 m³ =1000= 1000 L.

Step 1: Write the mathematical expression

Use the radius in meters, then convert m³ to liters:

Key Takeaways

  • 1A cylinder has two identical circular bases, and its volume is base area times height: V=B×hV = B \times h
  • 2Because the base is a circle, V=πr2hV = \pi r^2 h
  • 3The formula needs the radius: if you are given the diameter, divide it by 22 first
  • 4Give the answer in terms of π\pi (exact, like 90π90\pi cm³) or use π≈3.14\pi \approx 3.14 and round only at the end
  • 5To find the height, divide the volume by the base area: h=V÷(πr2)h = V \div (\pi r^2); to find the radius, find r2r^2 first and then take the square root
  • 611 cm³ =1= 1 mL, 10001000 cm³ =1= 1 L and 11 m³ =1000= 1000 L

Frequently Asked Questions

Because πr2\pi r^2 is the area of the circular base. The volume of any prism-like shape is base area times height, and a cylinder is like a prism with a circle as its base.
Because πr2\pi r^2 is the area of the circular base. The volume of any prism-like shape is base area times height, and a cylinder is like a prism with a circle as its base.
When the problem asks for an exact answer or says "in terms of π\pi". Otherwise use π≈3.14\pi \approx 3.14 (or the π\pi key on a calculator) and round to the place the problem asks for.
Nothing changes. The height is always the distance between the two circular bases, so for a pipe or a rolling pin lying down, the height is its length.
The volume becomes 4 times bigger, because the radius is squared: (2r)2=4r2(2r)^2 = 4r^2. Doubling the height only doubles the volume.

Glossary

Cylinder
A 3D shape with two identical, parallel circular bases joined by a curved surface
Base
One of the two circles at the ends of a cylinder; its area is B=πr2B = \pi r^2
Radius
The distance from the center of the base to its edge (rr)
Diameter
The distance across the base through the center; it is twice the radius (d=2rd = 2r)
Height
The distance between the two bases of the cylinder (hh)
Pi (π\pi)
The number you get when you divide a circle's circumference by its diameter, about 3.143.14

Formula Card

Volume of a cylinder

V=πr2hV = \pi r^2 h

Pi times the radius squared times the height

Base area times height

V=B×hV = B \times h, with B=πr2B = \pi r^2

The same rule as for prisms

Radius from diameter

r=d÷2r = d \div 2

Always halve the diameter before using the formula

Missing height

h=V÷(πr2)h = V \div (\pi r^2)

Divide the volume by the base area

Liters

1000 cm3=1 L1000 \text{ cm}^3 = 1 \text{ L}

Divide cm³ by 1000 to get liters ($1$ cm³ $= 1$ mL)

Today's challengeOne problem, three tries

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