Back to Lesson

Teacher Guide: 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.

Present LessonShare to ClassroomView Student Page

Use this lesson with your class

Free, no student accounts needed.

Share with students

Students open the lesson and practise with instant feedback.

Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Volume & Capacity. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Explain why the volume of a cylinder is base area times height
  • Calculate the volume of a cylinder with V=πr2hV = \pi r^2 h, given the radius or the diameter
  • Give volumes exactly in terms of π\pi and approximately with π≈3.14\pi \approx 3.14, rounding as stated
  • Find a missing height or radius when the volume is known
  • Convert cylinder volumes from cm³ to milliliters and liters in real situations
Prerequisites
  • • Area of a circle (A=πr2A = \pi r^2) and the link between radius and diameter
  • • Volume of rectangular prisms as base area times height
  • • Metric volume units (mL, L) and 10001000 mL =1= 1 L
  • • Squares and square roots of whole numbers
Discussion Starters
  • 1. How is a cylinder like a stack of coins? What does that tell you about its volume?
  • 2. Two glasses: one is twice as tall, the other twice as wide. Which one holds more? Why?
  • 3. Why might a math teacher prefer an answer like 90π90\pi cm³ over 282.6282.6 cm³?
  • 4. A can label says 330 mL. How could you check that with a ruler?
Common Misconceptions

Using the diameter in the formula because it is the number that is given

Remediation: Ask students to underline whether the problem says radius or diameter and to write r=…r = \ldots as the first line every time. Compare π×102\pi \times 10^2 and π×52\pi \times 5^2 to show the factor of 4.

Thinking that doubling the radius doubles the volume

Remediation: Calculate the volume for r=2r = 2 and r=4r = 4 with the same height (4πh4\pi h vs 16πh16\pi h). Let students fill two real cylinders with rice or water to see the difference.

Believing that cm³ and mL are different amounts that need a complicated conversion

Remediation: Show a 1 cm cube next to a 1 mL measuring spoon or syringe, then a 10 cm cube that holds exactly 1 liter.

Differentiation Ideas

For Struggling Students:

  • • Start from the base area: compute πr2\pi r^2 first, then multiply by the height as a separate step
  • • Use whole-number radii and heights and the value π≈3.14\pi \approx 3.14 only
  • • Give a checklist: radius or diameter? square it, multiply by height, multiply by 3.14, units

For On-Level Students:

  • • Mix problems with radius and diameter and ask for both exact and rounded answers
  • • Find a missing height from a given volume
  • • Convert tank and can volumes to liters and milliliters

For Advanced Students:

  • • Find a missing radius using square roots
  • • Investigate how the volume changes when the radius or height is scaled by 2 or 3
  • • Design a can that holds 500 mL and compare different radius and height choices
Standards Alignment
  • 8.G.C.9 (CCSS.MATH.CONTENT.8.G.C.9)

    Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems

  • HSG.GMD.A.3 (CCSS.MATH.CONTENT.HSG.GMD.A.3)

    Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems

Lesson Resources
  • visual3D Cylinder Viewer

    Students change the radius and height of a cylinder and watch the volume update

  • activityCan Measuring Lab

    Students measure real cans and glasses, calculate the volume and compare it with the label in mL

  • worksheetCylinder Volume Practice

    Problems with radius or diameter, exact answers in terms of π\pi, missing heights and liters

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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}.

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.

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

Why is there a πr2\pi r^2 in the formula?

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 should I leave the answer in terms of π\pi?

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.

What if the cylinder is lying on its side?

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.

What happens to the volume if I double the radius?

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)

More in This Topic

Introduction to Volume

Elementary20 min

Metric Volume Units (mL, L)

Elementary20 min

Volume of Rectangular Prisms

Intermediate25 min

Imperial Volume Units (cups, pints, gallons)

Elementary20 min