MTH1W Volume: Prisms, Pyramids, Cylinders and Cones: Notes and Practice

At a movie theatre you might see popcorn sold in a cylinder-shaped tub and in a cone-shaped cup with the same width across the top and the same height. They look as if they hold similar amounts, but the cone holds exactly one third of what the cylinder holds. The same is true of a pyramid and a box with the same base and height. Volume, the amount of space inside a 3-D object, decides how much a can, a tank, a silo or a bag of concrete holds, so knowing these formulas, and the one-third pattern that links them, helps you compare packages and plan real projects.

Prisms and cylinders: V = Bh

A prism has two matching parallel faces (the bases) joined by flat sides, and every slice parallel to the base is the same shape and size as the base. A cylinder is like a prism with a circular base. Think of building either one from thin layers stacked up to the height $h$ : each layer has the base's area $B$ , so the volume is the base area times the height. For a rectangular prism, $B = lw$ , so $V = lw h$ . For a triangular prism, $B$ is the triangle's area, $\frac{1}{2} \times base \times height of the triangle .$ For a cylinder, $B = π r^{2}$ , so $V = π r^{2} h$ .

Solid	Base area B	Volume
Rectangular prism	$lw$	$V = lw h$
Triangular prism	$\frac{1}{2} b h_{△}$	$V = \frac{1}{2} b h_{△} \times ℓ$ (prism length $ℓ$ )
Cylinder	$π r^{2}$	$V = π r^{2} h$
Square-based pyramid	$s^{2}$	$V = \frac{1}{3} s^{2} h$
Cone	$π r^{2}$	$V = \frac{1}{3} π r^{2} h$

Volume formulas built from base area times height

Fig 1A prism and a pyramid with the same square base and height, and a cylinder and a cone with the same radius and height, each labelled with its volume formula

Quick checkTry it before you read on

A triangular prism is 10 cm long. Its triangular base has a base of 6 cm and a height of 4 cm. What is its volume?

cm³

Quick checkTry it before you read on

A cylinder has a radius of 3 cm and a height of 10 cm. What is its volume, to the nearest tenth of a cubic centimetre?

cm³

VideoMath Antics explains volume as the area of the base times the height, the same idea as V = Bh for prisms and cylinders.Watch on YouTube

Pyramids and cones: one third

A pyramid has a polygon base and triangular sides that meet at a point, the apex. A cone has a circular base and a curved side that narrows to an apex. If you fill a hollow pyramid with water or sand and pour it into a hollow prism with the same base and the same height, it takes exactly three pourings to fill the prism. A cone and a cylinder with the same base and height behave the same way. So a pyramid or cone holds one third of the matching prism or cylinder.

V_{pyramid} = \frac{1}{3} B h V_{cone} = \frac{1}{3} π r^{2} h

Use the perpendicular height

The

h

in every volume formula is the perpendicular height: the straight-up distance from the base to the apex or top, measured at a right angle to the base. It is not the slant height, the distance measured along a sloping face. If you are given the slant height, use the Pythagorean theorem to find the perpendicular height first.

Fig 2A cone with radius 6 cm and slant height 10 cm drawn to scale; the perpendicular height, the radius and the slant height form a right triangle, so the height is 8 cm and the volume is 96π, about 301.6 cubic centimetres

Quick checkTry it before you read on

A cone holds 150 mL. A cylinder with the same radius and the same height holds how much?

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A cone has a slant height of 13 cm and a radius of 5 cm. Which height goes into the volume formula?

VideoA pouring demonstration with a cone and a cylinder of the same base and height, like the experiment described in this section.Watch on YouTube

Keeping units consistent

Every length in a volume formula must be in the same unit before you multiply. A tank with a radius of 60 cm and a height of 1.5 m must be worked either all in centimetres (60 cm and 150 cm) or all in metres (0.6 m and 1.5 m). Lengths in cm give a volume in $cm^{3}$ ; lengths in m give $m^{3}$ . Capacity links to volume: $1 cm^{3}$ holds exactly $1 mL$ , so dividing $cm^{3}$ by 1000 gives litres.

Fig 3A tank with radius 60 cm and height 1.5 m: working all in centimetres or all in metres both give about 1696 L, while mixing centimetres and metres gives a meaningless number

Volume	Capacity
$1 cm^{3}$	1 mL
$1000 cm^{3}$	1 L
$1 m^{3} = 1000000 cm^{3}$	1000 L

Volume and capacity links

Working backwards

If you know the volume and need a height, substitute what you know and solve. For a cone with

V = 300 cm^{3}

and

r = 5

cm:

300 = \frac{1}{3} π (5)^{2} h

, so

h = \frac{3 \times 300}{25 π} \approx 11.5

cm (rounded to one decimal place).

Quick checkTry it before you read on

A box is 50 cm long, 40 cm wide and 30 cm tall. How many litres does it hold?

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A cylinder has a radius of 20 cm and a height of 2 m. Which calculation keeps the units consistent?

Volume: Prisms, Pyramids, Cylinders and Cones

Key things to remember

Prisms and cylinders: V = Bh

Pyramids and cones: one third

Keeping units consistent

Worked examples

Where marks get lost

Flashcards 8

Practice