Year 11 General Shape And Measurement

Volume Of Other Solids

Q: What is the formula for the volume of a cone?

V equals one third times pi r squared times h, where r is the radius of the circular base and h is the perpendicular height from base to apex.

Q: What is the formula for the volume of a pyramid?

V equals one third times the base area times the perpendicular height. The base can be any shape — square, rectangle, or triangle.

Q: What is the formula for the volume of a sphere?

V equals four thirds pi r cubed. Note that r is cubed (to the power 3), not squared.

Q: What is the formula for the volume of a hemisphere?

V equals two thirds pi r cubed. It is half the volume of the full sphere.

Q: Why is the volume of a cone one third of a cylinder?

It is a geometric property: any cone or pyramid takes up exactly one third of the space of the prism or cylinder it would fit inside, assuming they share the same base and height.

12 practice questions 0 video lessons Theory + worked examples

Theory

Beyond prisms, three other shapes appear regularly: cones, pyramids, and spheres. Cones and pyramids have a single apex and their volume is exactly one-third of the matching prism. A sphere's volume is $\frac{4}{3} π r^{3}$ — note the $r^{3}$ , not $r^{2}$ .

Beyond prisms, three other 3D shapes appear regularly: cones, pyramids, and spheres. Cones and pyramids each have a single apex (point) and one base.

The volume of a cone or pyramid is exactly one-third of the matching prism — a cone is one-third of the cylinder with the same base and height; a square pyramid is one-third of the cuboid that would fit around it.

A sphere is the 3D analogue of a circle: every point on its surface is the same distance $r$ from the centre. Its volume formula is $V = \frac{4}{3} π r^{3}$ — note that $r$ is cubed, not squared. A hemisphere is half a sphere: $V = \frac{2}{3} π r^{3}$ .

Composite solids (silos, capsules, ice-cream cones with a scoop) are built from simple solids stuck together. Compute each part's volume separately and add.

Three new shapes, three new formulas

Same idea: pyramid = ⅓ of cuboid

Pyramid (any base):

V = \frac{1}{3} \cdot A_{base} \cdot h

V = \frac{1}{3} A_{base} h

Cone (circular base, radius $r$ ):

V = \frac{1}{3} \cdot π r^{2} \cdot h

V = \frac{1}{3} π r^{2} h

Sphere (radius $r$ ):

V = \frac{4}{3} π r^{3}

V = \frac{4}{3} π r^{3}

Hemisphere (half a sphere):

V = \frac{2}{3} π r^{3}

At a glance:

Solid	Volume
Pyramid	$\frac{1}{3} A_{base} \cdot h$
Cone	$\frac{1}{3} π r^{2} h$
Sphere	$\frac{4}{3} π r^{3}$
Hemisphere	$\frac{2}{3} π r^{3}$

Capacity conversions:

1,000

cm³

= 1

1

m³

= 1,000

How to find the volume of any of these solids

Identify the solid — pyramid, cone, sphere, hemisphere, or a composite.
Use the perpendicular height, not slant height, for cone and pyramid formulas.
For a cone, use the radius (halve the diameter if needed) in $V = \frac{1}{3} π r^{2} h$ .
For a sphere, use $V = \frac{4}{3} π r^{3}$ . The exponent is $3$ , not $2$ .
For a composite solid, split into parts, compute each volume, and add.

Example 1 — Cone

A cone has radius

6

cm and perpendicular height

10

cm. Find its volume (2 dp).

Solution

Use $V = \frac{1}{3} π r^{2} h$ with $r = 6$ , $h = 10$ .

$V$	$=$	$\frac{1}{3} \times π \times 6^{2} \times 10$
$V$	$=$	$\frac{1}{3} \times 360 π = 120 π$
$V$	$\approx$	$376.99$ cm³

V \approx 376.99

The volume is about $376.99$ cm³.

Example 2 — Square pyramid

A square pyramid has a base

8

m by

8

m and a vertical height of

9

m. Find its volume.

Solution

Base area first, then apply $V = \frac{1}{3} A_{base} h$ .

$A_{base}$	$=$	$8 \times 8 = 64$ m²
$V$	$=$	$\frac{1}{3} \times 64 \times 9$
$V$	$=$	$192$ m³

V = 192

The volume is $192$ m³.

Example 3 — Sphere

A sphere has a diameter of

12

cm. Find its volume (2 dp).

Solution

Halve the diameter to get $r$ , then apply $V = \frac{4}{3} π r^{3}$ .

$r$	$=$	$12 \div 2 = 6$
$V$	$=$	$\frac{4}{3} \times π \times 6^{3}$
$V$	$=$	$\frac{4}{3} \times 216 π = 288 π$
$V$	$\approx$	$904.78$ cm³

V \approx 904.78

The volume is about $904.78$ cm³.

Example 4 — Composite silo

A grain silo has a cylindrical body (radius

2

m, height

5

m) and a conical top (same radius, height

3

m). Find the total volume (2 dp).

Solution

Compute each part, then add.

$V_{cyl}$	$=$	$π \times 4 \times 5 = 20 π$
$V_{cone}$	$=$	$\frac{1}{3} \times 4 π \times 3 = 4 π$
$V_{tot}$	$=$	$20 π + 4 π = 24 π$
$V_{tot}$	$\approx$	$75.40$ m³

V \approx 75.40

The total volume is about $75.40$ m³.

Common pitfalls

Slant height instead of perpendicular height. The

h

in cone and pyramid formulas is the vertical drop from the apex to the base — not the slant along the surface. The slant is longer and inflates the volume.

Using diameter for a cone. The cone formula is

V = \frac{1}{3} π r^{2} h

. If given the diameter, halve it before substituting.

Using $r^{2}$ instead of $r^{3}$ for a sphere. The formula is

V = \frac{4}{3} π r^{3}

. Forgetting the cube is the most common sphere error.

Forgetting the $\frac{1}{3}$ . Cones and pyramids share the

\frac{1}{3}

factor — drop it and your answer is three times too big.

Frequently asked questions

What is the formula for the volume of a cone?

$V = \frac{1}{3} π r^{2} h$ , where $r$ is the base radius and $h$ is the perpendicular height.

What is the formula for the volume of a pyramid?

$V = \frac{1}{3} \cdot A_{base} \cdot h$ . Works for any base shape — square, rectangle, or triangle.

What is the formula for the volume of a sphere?

$V = \frac{4}{3} π r^{3}$ . Note that $r$ is cubed (to the power $3$ ), not squared.

What is the formula for the volume of a hemisphere?

$V = \frac{2}{3} π r^{3}$ — half the volume of the full sphere.

Why is the volume of a cone one third of a cylinder?

It's a geometric property: any cone or pyramid takes up exactly one-third of the space of the prism or cylinder it fits inside, when they share the same base and height.

How do you find the volume of a composite solid?

Split it into simple solids (prisms, cones, hemispheres, etc.), compute each volume separately, and add them up.