E. Geometry and Measurement · Geometry and Measurement: Circles, Triangles, Measurement Systems, Pythagoras and Volume

The Volume of Prisms and Pyramids

E1.6 E1.5 E1.3 A1About 70 minutes

Learning goals

I can find the volume of any right prism with V = (area of base) × height.
I can explain why the volume of a pyramid is one-third of the volume of a prism with the same base and height.
I can find the volume of a pyramid, and work backwards to find a missing height or base.
I can solve volume problems with different units, including mL, L, cm³ and m³.

Words to know

prism
A 3D object with two identical, parallel bases joined by rectangular faces. It is named after its base.

pyramid
A 3D object with one polygon base and triangular faces that meet at the apex.

base area (B)
The area of the base of a prism or pyramid.

height (h)
The perpendicular distance between the bases of a prism, or from the base of a pyramid to its apex.

volume
The amount of space inside a 3D object, in cubic units. For liquids we use capacity units: 1 cm³ = 1 mL.

A hook: the pyramid-shaped snack box

A company sells crackers in a box shaped like a square pyramid. Its base is the same as a regular box and it is just as tall. Does it hold about the same amount? Half? Less?

It holds exactly one-third as much. Today you will see why, and you will find volumes of prisms and pyramids in many units.

Remember from Grade 7 and 8

The volume of a prism is V = B × h, where B is the area of the base and h is the height (the distance between the bases). This works for every right prism: rectangular, triangular, pentagonal, and even house-shaped.

A triangular prism. First find the area of the triangle (the base), then multiply by the length between the two triangular ends.

A prism and a pyramid with the same square base and the same height. The pyramid's volume is one-third of the prism's.

Why one-third?

Experiment: fill a hollow pyramid with rice or sand and pour it into a hollow prism with the same base and height. It takes 3 pyramids to fill the prism.
A cube cut into three: a cube can be cut into 3 identical square pyramids, each with a face of the cube as its base and the same opposite corner as its apex. Each one has base B = s² and height h = s, and together they make the cube, s³. So each pyramid is ⅓ × B × h.

So, for any pyramid: V = ⅓ × B × h.

Three pyramids fill one prism with the same base and height.

Worked example

Example 1: Prism and pyramid

A square pyramid has a base 6 cm on each side and a height of 10 cm. Find its volume, and the volume of a prism with the same base and height.

B = 6 × 6 = 36 cm².
Prism: V = 36 × 10 = 360 cm³.
Pyramid: V = ⅓ × 36 × 10 = 120 cm³.

Pyramid 120 cm³; prism 360 cm³

Worked example

Example 2: Triangular prism

A chocolate bar box is a triangular prism. Each triangular end has a base of 8 cm and a height of 6 cm, and the box is 25 cm long. Find its volume in cm³ and mL.

B = ½ × 8 × 6 = 24 cm².
V = 24 × 25 = 600 cm³.
1 cm³ = 1 mL, so 600 cm³ = 600 mL.

600 cm³ = 600 mL

Worked example

Example 3: Using a slant height

A square pyramid has a base of 10 m and a slant height of 13 m. Find its volume.

We need the height. The height, half the base (5 m) and the slant height make a right triangle.
h = √(13² − 5²) = √144 = 12 m.
V = ⅓ × 10² × 12 = ⅓ × 1200 = 400 m³.

400 m³

Worked example

Example 4: Missing height

A square pyramid has a volume of 96 cm³ and a base 6 cm on each side. How tall is it?

V = ⅓Bh, so 96 = ⅓ × 36 × h = 12h.
h = 96 ÷ 12 = 8 cm.
Shortcut: h = 3V ÷ B = 288 ÷ 36 = 8 cm.

8 cm

Worked example

Example 5: A skylight in litres

A glass skylight is a square pyramid with a base of 2.4 m and a height of 1.5 m. How many litres of air are inside it?

B = 2.4² = 5.76 m².
V = ⅓ × 5.76 × 1.5 = 2.88 m³.
1 m³ = 1000 L, so 2.88 m³ = 2880 L.

2880 L

Worked example

Example 6: A house-shaped prism

A shed has the house-shaped front from the last lesson (area 52 m²) and is 10 m deep. Find its volume.

The shed is a prism whose base is the front: B = 52 m².
V = 52 × 10 = 520 m³.

520 m³

Watch out!

B is an area, not a side length. For a square base of 6 cm, B = 36 cm², not 6.
Use the height, not the slant height, in V = ⅓Bh. Find the height with Pythagoras if needed.
Make the units match before you calculate (all cm or all m), and remember 1 m³ = 1000 L and 1 cm³ = 1 mL.

Hands-on activity

The rice test

You need: thin cardboard or heavy paper, a ruler, tape, dry rice, and scissors (a grown-up should help with cutting).

Open box (prism): make a 6 cm × 6 cm square base with four 6 cm × 6 cm sides. Tape it into an open cube.
Open pyramid: draw four isosceles triangles, each with a 6 cm base and two equal sides of 7.3 cm. Tape them together into a pyramid with no base. Its height is then about 6 cm, the same as the box. (Check with Pythagoras: from the centre of the base to a corner is √(3² + 3²) ≈ 4.24 cm, so each sloping edge is √(6² + 4.24²) = √54 ≈ 7.3 cm.)
Turn the pyramid upside down, fill it level with rice, and pour it into the box. Count how many pyramids fill the box.

What to look for: about 3 pyramids fill the box. Calculation check: box 6 × 6 × 6 = 216 cm³, pyramid ⅓ × 36 × 6 = 72 cm³, and 3 × 72 = 216.

Just for fun

A pyramid walks into a bakery and asks for one-third of a cake. "Same base, same height," it explains. "It's only fair."

Your turn

Find the volume of a rectangular prism 4 cm × 5 cm × 10 cm.

A square pyramid has a base of 9 cm and a height of 12 cm. Find its volume.

A triangular prism has a triangular base with base 6 cm and height 4 cm, and a length of 15 cm. Find its volume.

A prism has a volume of 270 cm³. What is the volume of a pyramid with the same base and height?

A pyramid has a volume of 200 cm³ and a base area of 40 cm². Find its height.

A rectangular pyramid has a base 8 m by 5 m and a height of 9 m. Find its volume.

Convert 2.5 m³ to litres.

More practice: warm-up

What is the volume formula for a prism?

What fraction of the matching prism is a pyramid?

1 cm³ = ? mL

A cube has sides of 3 cm. Its volume?

1 L = ? cm³

More practice: on your own

A square pyramid has a base of 16 m and a slant height of 17 m. Find its volume.

A tent is a triangular prism. The front is a triangle with base 2.4 m and height 1.6 m, and the tent is 3 m long. Find the volume of air inside.

A square pyramid must hold 1000 cm³ and have a base of 15 cm. How tall must it be, to one decimal?

A milk carton's main part is a prism 7 cm × 7 cm × 20.5 cm. About how many litres is that?

A grain hopper is an upside-down square pyramid with a top 1.2 m wide and a depth of 0.9 m. How many litres does it hold?

Challenge

The Great Pyramid of Giza originally had a square base about 230 m on each side and a height of about 146 m. Estimate its volume.

A tower is a square prism 4 m × 4 m × 6 m with a square pyramid roof of height 3 m on top. Find the total volume.

A cube with side s is cut into 3 identical pyramids, each using one face as its base and the same corner as its apex. Show that the formula V = ⅓Bh gives the right total.

Check your understanding

Quiz

A prism has a base area of 30 cm² and a height of 8 cm. Its volume is:

A pyramid and a prism have the same base and height. The prism's volume is 600 cm³. The pyramid's is:

A square pyramid has a base of 3 m and a height of 4 m. Its volume is:

A pyramid has V = 150 cm³ and B = 30 cm². Its height is:

Which length belongs in V = ⅓Bh for a pyramid?

A container holds 0.75 m³. How many litres is that?

For parents and teachers

Big idea: E1.6 asks students to solve problems using the relationship between the volumes of prisms and pyramids (and, next lesson, cylinders and cones), with various units. The key relationship: a pyramid holds one-third of a prism with the same base and height.

Common difficulties:

Using a side length instead of the base area.
Using the slant height instead of the height. Use the Pythagorean theorem from the last lesson to find h.
Unit conversions: 1 m³ = 1000 L (not 100).

How to help: The rice test makes the one-third rule unforgettable; supervise the cutting. Check package volumes at home (in mL) against measured dimensions.