Grade by Grade
E. Spatial Sense · Spatial Sense: 3D Objects, Transformations, Circles, Surface Area and Volume

Volume of Prisms and Cylinders

E2.7 A1About 50 minutes
Learning goals
  • I can explain why the volume of a prism or cylinder is the area of its base times its height.
  • I can find the volume of rectangular prisms, triangular prisms and cylinders.
  • I can find a missing height or base area when I know the volume.
  • I can connect volume to capacity in real problems.

Words to know

base area (B)
The area of the base of a prism or cylinder.
height (h)
The distance between the two bases of a prism or cylinder, measured straight across (at a right angle to the bases).
volume
The amount of space an object takes up, measured in cubic units.
V = B × h
The volume formula for every prism and cylinder: volume = area of base × height.

Remember from this unit

In the volume and capacity lesson you found the volume of a box by thinking about layers of cubes: length × width gives the cubes in one layer, and the height tells you how many layers. You also learned 1 cm³ = 1 mL, and how to find the area of a circle. Now you'll put it all together.

Let's start with a stack of coins

Stack 20 identical coins. Each coin is a thin cylinder. The stack is a taller cylinder. Its volume is just the volume of one coin times the number of coins. Every prism and every cylinder can be thought of as a stack of identical layers, each one the shape of the base.

Why V = B × h

Let B be the area of the base.

  • A layer 1 cm thick has a volume of B × 1 cubic centimetres.
  • If the object is h cm tall, it is h layers: V = B × h.

This works for every prism and cylinder. Only the base area changes:

  • Rectangular prism: B = length × width, so V = l × w × h.
  • Triangular prism: B = base × height of the triangle ÷ 2.
  • Cylinder: B = π × r × r, so V = π × r × r × h.
base: area BEach layer is 1 unit thick.Volume of 1 layer = B × 1The object is h layers tall.Volume = B × h
A cylinder sliced into layers. Each layer is a copy of the base, so the volume is the base area B times the number of layers, the height h.
Worked example

Example 1: Rectangular prism

Find the volume of a box 8 cm long, 5 cm wide and 4 cm tall.

  1. B = 8 × 5 = 40 cm².
  2. V = B × h = 40 × 4 = 160 cm³.

160 cm³

Worked example

Example 2: Triangular prism

A chocolate box is a triangular prism. The triangle has a base of 6 cm and a height of 4 cm. The box is 10 cm long. Find its volume.

  1. B = triangle area = 6 × 4 ÷ 2 = 12 cm².
  2. The prism's height (its length between the two triangles) is 10 cm.
  3. V = 12 × 10 = 120 cm³.

120 cm³. (Careful: the triangle's height and the prism's height are different measurements.)

4 cm6 cm10 cmB = 6 × 4 ÷ 2= 12 cm²h (prism) = 10 cmV = B × h= 12 × 10 = 120 cm³
The chocolate box from Example 2. The 4 cm height belongs to the triangle (it helps find B); the 10 cm length between the two triangles is the prism's height h.
Worked example

Example 3: Cylinder

A can has a radius of 4 cm and a height of 10 cm. Find its volume and its capacity in mL.

  1. B = 3.14 × 4 × 4 = 50.24 cm².
  2. V = 50.24 × 10 = 502.4 cm³.
  3. 1 cm³ = 1 mL, so it holds about 502 mL (if the walls are very thin).

About 502.4 cm³, or about 502 mL.

Worked example

Example 4: Missing height

A prism has a volume of 900 cm³ and a base area of 60 cm². How tall is it?

  1. V = B × h, so h = V ÷ B.
  2. h = 900 ÷ 60 = 15 cm.
  3. Check: 60 × 15 = 900 ✓

15 cm

Worked example

Example 5: Missing base

A cylinder has a volume of 1570 cm³ and a height of 20 cm. Find the base area and the radius.

  1. B = V ÷ h = 1570 ÷ 20 = 78.5 cm².
  2. B = π × r × r, so r × r = 78.5 ÷ 3.14 = 25.
  3. 5 × 5 = 25, so r = 5 cm.

Base area 78.5 cm², radius 5 cm.

Know two, find the third

V = B × h connects three measurements. If you know any two, you can find the third:

  • V = B × h
  • h = V ÷ B
  • B = V ÷ h

And always write volume in cubic units (cm³, m³).

Hands-on activity

Check a can's label

You need: a sealed can of soup or vegetables (keep it sealed: never open a can or touch cut can edges) with the capacity (mL) printed on it, a ruler, string.

  1. Measure the can's height and its circumference (with string). Find the diameter: C ÷ 3.14. Then find the radius.
  2. Calculate the volume: 3.14 × r × r × h.
  3. Compare with the mL on the label. Is your answer a bit bigger? (It should be, because cans aren't filled right to the top and the rims take up space.)
  4. Bonus: find a cereal box. Measure it and calculate its volume. Would 1 L of water fit inside?

What to look for: your volume in cm³ is close to, and a little more than, the mL on the label.

Just for fun

A famous math joke: a pizza with radius z and thickness a has a volume of π × z × z × a. Read it out loud: pi z z a!

Your turn

1

A prism has a base area of 25 cm² and a height of 8 cm. Find its volume.

2

Find the volume of a box 6 cm by 4 cm by 5 cm.

3

Find the volume of a cylinder with r = 2 cm and h = 10 cm.

4

A triangular prism has a triangle with base 4 cm and height 3 cm. The prism is 9 cm long. Find its volume.

5

A prism has V = 240 cm³ and B = 30 cm². Find its height.

6

A prism has V = 500 cm³ and h = 20 cm. Find its base area.

7

Find the volume of a cylinder with a diameter of 10 cm and a height of 6 cm.

More practice: warm-up

1

B = 10 cm², h = 3 cm. V = ?

2

Find the volume of a cube with 3 cm edges.

3

Find the volume of a cylinder with r = 1 cm and h = 1 cm.

4

What unit is used for volume: cm, cm² or cm³?

5

B = 12 cm², V = 36 cm³. What is h?

More practice: on your own

1

A juice can has r = 3 cm and h = 12 cm. About how many mL does it hold?

2

A tent is a triangular prism. The triangle has a base of 2 m and a height of 1.5 m. The tent is 3 m long. How much space is inside?

3

A round wading pool has r = 3 m and is filled 1 m deep. How many litres of water is that? (1 m³ = 1000 L)

4

A box has a base 15 cm by 10 cm and holds exactly 1.2 L. How tall is it inside?

5

Cylinder A has r = 2 cm and h = 8 cm. Cylinder B has r = 4 cm and h = 2 cm. Which holds more?

Challenge

1

You want to design a cylinder that holds exactly 1 L, with a base area of 50 cm². How tall must it be?

2

A cylinder has a volume of 1256 cm³ and a height of 4 cm. Find its radius.

3

A cube-shaped box has 10 cm edges. A cylinder with a diameter of 10 cm and a height of 10 cm fits exactly inside it. How much space in the box is not filled by the cylinder?

Check your understanding

Quiz

1.

Which formula gives the volume of any prism or cylinder?

2.

A prism has B = 20 cm² and h = 7 cm. What is its volume?

3.

What is the volume of a cylinder with r = 10 cm and h = 2 cm?

4.

A prism has V = 360 cm³ and B = 40 cm². What is its height?

5.

A triangular prism has a triangle with base 6 cm and height 5 cm, and a length of 10 cm. What is its volume?

6.

A container has a volume of 500 cm³ inside. What is its capacity?

For parents and teachers

Big idea: E2.7 asks students to show that the volume of a prism or cylinder is the area of its base times its height, and to use this to find the base area, volume or height when two of them are known.

How to help at home: Stack coins, crackers or sticky notes to show layers. Measure a can and compare your calculation with the mL on the label. Ask, "If you know the volume and the height, how can you find the base area?"

Common difficulties:

  • Mixing up the triangle's height and the prism's height in triangular prisms.
  • Forgetting to divide by 2 for a triangular base.
  • Using the diameter instead of the radius for a cylinder's base.
  • Writing cm² instead of cm³.

This lesson builds A1 skills: connecting ideas (area, layers, capacity) and using one formula flexibly.