Spheres and Cones

What you'll learn

Work with cones: 3D shapes with a circular base and a point.
Work with spheres: perfectly round 3D shapes, and hemispheres: half-spheres.
Calculate volume — the space inside a 3D shape.
Calculate surface area — the area on the outside of a 3D shape.

Key measurements in diagrams

Definition

The letters in the formulas

The radius, $r$ , goes from the centre of a circle or sphere to its edge.
The diameter, $d$ , goes all the way across a circle through the centre, so $d = 2 r$ and $r=d2r=\frac{d}{2}$ .
The vertical height, $h$ , is the straight up-and-down height of a cone.
The slant height, $l$ , is the length down the sloping side of a cone.

Key measurements on a cone: radius, diameter, vertical height and slant height.

Key Idea

Find the radius first

Most sphere and cone formulas use the radius, not the diameter. If you are given a diameter, halve it before using a formula.

Example

Reading a cone diagram

A cone has base diameter 14 cm and vertical height 20 cm. Find the values of $r$ and $h$ .

Cone with diameter 14 cm and vertical height 20 cm, so the radius is half the diameter.

The vertical height is already given.
$h = 20$
Halve the diameter to find the radius.
$r=142=7r=\frac{14}{2}=7$
Use $r = 7$ and $h = 20$ in any cone formula.

Volume: space inside

For a cone:

Vcone=13πr2hV_{\text{cone}}=\frac{1}{3}\pi r^2h

For a sphere:

Vsphere=43πr3V_{\text{sphere}}=\frac{4}{3}\pi r^3

A hemisphere is half a sphere, so its volume is half the sphere volume.

Example

Volume of a cone

A cone has height 18 cm and base diameter 10 cm. Work out its volume to 3 significant figures.

For cone volume, use the vertical height and the radius found from the diameter.

Find the radius.
$r=102=5r=\frac{10}{2}=5$
Substitute into the cone volume formula.
$V=13π×52×18V=\frac{1}{3}\pi \times 5^2 \times 18$
Calculate.
$V=150π≈471.239V=150\pi \approx 471.239$
The volume is 471 cm³ to 3 significant figures.

Tip

Rounding

Do all calculator work first, then round at the end. Rounding too early can make your final answer less accurate.

Surface area: outside area

Definition

Surface area words

Curved surface area means only the curved outside part.
Total surface area means every outside part, including flat circular bases.

For a sphere:

Asphere=4πr2A_{\text{sphere}}=4\pi r^2

For a solid hemisphere, take half the sphere surface area, then add the flat circular base:

hemisphere=2πr2+πr2=3πr2A_{\text{total hemisphere}}=2\pi r^2+\pi r^2=3\pi r^2

Example

Total surface area of a hemisphere

A solid hemisphere has radius 7 cm. Work out its total surface area in terms of $π\pi$ .

A solid hemisphere’s total surface area includes the curved half-sphere and the flat circular base.

Use the total surface area formula for a solid hemisphere.
$A=3πr2A=3\pi r^2$
Substitute $r = 7$ .
$A=3π×72A=3\pi \times 7^2$
Simplify.
$A=147πA=147\pi$
The total surface area is $147π147\pi$ cm².

For a cone:

cone=πrlA_{\text{curved cone}}=\pi rl

So for a solid cone:

cone=πrl+πr2A_{\text{total cone}}=\pi rl+\pi r^2

Common Mistake

Forgetting the flat circle

For a solid cone or solid hemisphere, total surface area includes the flat circular face. Curved surface area alone is not the total surface area.

Finding the slant height

Sometimes a cone question gives the vertical height, not the slant height.

Definition

Pythagoras' theorem

In a right-angled triangle, the square of the longest side equals the sum of the squares of the other two sides. In a cone cross-section, $l$ is the longest side, so $l^2=h^2+r^2$ .

A vertical cross-section of a cone gives a right-angled triangle with legs h and r and hypotenuse l.

Example

Cone surface area when height is given

A solid cone has vertical height 8 cm and base diameter 12 cm. Work out its total surface area in terms of $π\pi$ .

Use the diameter to get the radius, then use the right triangle to find the slant height before calculating total surface area.

Find the radius.
$r=122=6r=\frac{12}{2}=6$
Use Pythagoras to find the slant height.
$l^2=8^2+6^2=100$
Square root to find $l$ .
$l = 10$
Add curved area and base area.
$A=π×6×10+π×62=96πA=\pi \times 6 \times 10+\pi \times 6^2=96\pi$
The total surface area is $96π96\pi$ cm².

Combined shapes

Definition

Compound solid

A compound solid is a 3D shape made by joining simpler 3D shapes. For volume, add the volumes of the separate parts.

Example

Cone on a hemisphere

A shape is made from a cone on top of a hemisphere. Both have diameter 8 cm. The cone has height 9 cm. Find the total volume in terms of $π\pi$ .

The compound solid is a cone joined to a hemisphere with the same circular diameter.

Find the shared radius.
$r=82=4r=\frac{8}{2}=4$
Find the cone volume.
$Vcone=13π×42×9=48πV_{\text{cone}}=\frac{1}{3}\pi \times 4^2 \times 9=48\pi$
Find the hemisphere volume.
$Vhemisphere=12×43π×43=1283πV_{\text{hemisphere}}=\frac{1}{2}\times \frac{4}{3}\pi \times 4^3=\frac{128}{3}\pi$
Add the two volumes.
$Vtotal=48π+1283π=2723πV_{\text{total}}=48\pi+\frac{128}{3}\pi=\frac{272}{3}\pi$
The total volume is $2723π\frac{272}{3}\pi$ cm³.

Tip

In terms of pi

If the question asks for an answer in terms of $π\pi$ , leave $π\pi$ in your answer instead of converting to a decimal.

Working backwards from volume

Sometimes you know the volume and need to find the radius. A cube root undoes cubing.

Example

Sphere and cube with equal volume

A cube has side length 6 cm. A sphere has the same volume as the cube. Find the radius of the sphere to 3 significant figures.

Find the cube volume.
$Vcube=63=216V_{\text{cube}}=6^3=216$
Set the sphere volume equal to 216.
$43πr3=216\frac{4}{3}\pi r^3=216$
Rearrange to find $r^3$ .
$r3=216×34π=162πr^3=\frac{216\times 3}{4\pi}=\frac{162}{\pi}$
Cube root to find $r$ .
$r=162π3≈3.72r=\sqrt[3]{\frac{162}{\pi}}\approx 3.72$
The radius is 3.72 cm.

Water level rise

When a solid sinks fully under water, it pushes up the water by its own volume.

base\text{rise}=\frac{\text{volume of solid}}{\text{area of container base}}

Example

Rise in water level

A rectangular container has base 12 cm by 10 cm. A metal sphere of radius 3 cm sinks fully under the water. Find the rise in water level to 3 significant figures.

A fully submerged sphere displaces its own volume, causing the water level in the rectangular container to rise.

Find the area of the base of the container.
$Abase=12×10=120A_{\text{base}}=12\times 10=120$
Find the volume of the sphere.
$V=43π×33=36πV=\frac{4}{3}\pi \times 3^3=36\pi$
Divide by the base area.
$rise=36π120≈0.942\text{rise}=\frac{36\pi}{120}\approx 0.942$
The water rises by 0.942 cm.

Exam technique

In the exam

First mark the radius; if you see a diameter, halve it.
Check whether the question wants volume, curved surface area, or total surface area.
If a cone surface area question gives vertical height, use Pythagoras to find the slant height.
Leave answers in terms of $π\pi$ when asked; otherwise round only at the final step.

Self review

Check yourself

What extra part must you include for the total surface area of a solid hemisphere?
Which cone formula uses the slant height $l$ ?
How do you turn a displaced volume into a rise in water level?

Spheres and Cones

What you'll learn

Work with cones: 3D shapes with a circular base and a point.
Work with spheres: perfectly round 3D shapes, and hemispheres: half-spheres.
Calculate volume — the space inside a 3D shape.
Calculate surface area — the area on the outside of a 3D shape.

Key measurements in diagrams

Definition

The letters in the formulas

The radius, $r$ , goes from the centre of a circle or sphere to its edge.
The diameter, $d$ , goes all the way across a circle through the centre, so $d = 2 r$ and $r=d2r=\frac{d}{2}$ .
The vertical height, $h$ , is the straight up-and-down height of a cone.
The slant height, $l$ , is the length down the sloping side of a cone.

Key measurements on a cone: radius, diameter, vertical height and slant height.

Key Idea

Find the radius first

Most sphere and cone formulas use the radius, not the diameter. If you are given a diameter, halve it before using a formula.

Example

Reading a cone diagram

A cone has base diameter 14 cm and vertical height 20 cm. Find the values of $r$ and $h$ .

Cone with diameter 14 cm and vertical height 20 cm, so the radius is half the diameter.

The vertical height is already given.
$h = 20$
Halve the diameter to find the radius.
$r=142=7r=\frac{14}{2}=7$
Use $r = 7$ and $h = 20$ in any cone formula.

Volume: space inside

For a cone:

Vcone=13πr2hV_{\text{cone}}=\frac{1}{3}\pi r^2h

For a sphere:

Vsphere=43πr3V_{\text{sphere}}=\frac{4}{3}\pi r^3

A hemisphere is half a sphere, so its volume is half the sphere volume.

Example

Volume of a cone

A cone has height 18 cm and base diameter 10 cm. Work out its volume to 3 significant figures.

For cone volume, use the vertical height and the radius found from the diameter.

Find the radius.
$r=102=5r=\frac{10}{2}=5$
Substitute into the cone volume formula.
$V=13π×52×18V=\frac{1}{3}\pi \times 5^2 \times 18$
Calculate.
$V=150π≈471.239V=150\pi \approx 471.239$
The volume is 471 cm³ to 3 significant figures.

Tip

Rounding

Do all calculator work first, then round at the end. Rounding too early can make your final answer less accurate.

Surface area: outside area

Definition

Surface area words

Curved surface area means only the curved outside part.
Total surface area means every outside part, including flat circular bases.

For a sphere:

Asphere=4πr2A_{\text{sphere}}=4\pi r^2

For a solid hemisphere, take half the sphere surface area, then add the flat circular base:

hemisphere=2πr2+πr2=3πr2A_{\text{total hemisphere}}=2\pi r^2+\pi r^2=3\pi r^2

Example

Total surface area of a hemisphere

A solid hemisphere has radius 7 cm. Work out its total surface area in terms of $π\pi$ .

A solid hemisphere’s total surface area includes the curved half-sphere and the flat circular base.

Use the total surface area formula for a solid hemisphere.
$A=3πr2A=3\pi r^2$
Substitute $r = 7$ .
$A=3π×72A=3\pi \times 7^2$
Simplify.
$A=147πA=147\pi$
The total surface area is $147π147\pi$ cm².

For a cone:

cone=πrlA_{\text{curved cone}}=\pi rl

So for a solid cone:

cone=πrl+πr2A_{\text{total cone}}=\pi rl+\pi r^2

Common Mistake

Forgetting the flat circle

For a solid cone or solid hemisphere, total surface area includes the flat circular face. Curved surface area alone is not the total surface area.

Finding the slant height

Sometimes a cone question gives the vertical height, not the slant height.

Definition

Pythagoras' theorem

In a right-angled triangle, the square of the longest side equals the sum of the squares of the other two sides. In a cone cross-section, $l$ is the longest side, so $l^2=h^2+r^2$ .

A vertical cross-section of a cone gives a right-angled triangle with legs h and r and hypotenuse l.

Example

Cone surface area when height is given

A solid cone has vertical height 8 cm and base diameter 12 cm. Work out its total surface area in terms of $π\pi$ .

Use the diameter to get the radius, then use the right triangle to find the slant height before calculating total surface area.

Find the radius.
$r=122=6r=\frac{12}{2}=6$
Use Pythagoras to find the slant height.
$l^2=8^2+6^2=100$
Square root to find $l$ .
$l = 10$
Add curved area and base area.
$A=π×6×10+π×62=96πA=\pi \times 6 \times 10+\pi \times 6^2=96\pi$
The total surface area is $96π96\pi$ cm².

Combined shapes

Definition

Compound solid

A compound solid is a 3D shape made by joining simpler 3D shapes. For volume, add the volumes of the separate parts.

Example

Cone on a hemisphere

A shape is made from a cone on top of a hemisphere. Both have diameter 8 cm. The cone has height 9 cm. Find the total volume in terms of $π\pi$ .

The compound solid is a cone joined to a hemisphere with the same circular diameter.

Find the shared radius.
$r=82=4r=\frac{8}{2}=4$
Find the cone volume.
$Vcone=13π×42×9=48πV_{\text{cone}}=\frac{1}{3}\pi \times 4^2 \times 9=48\pi$
Find the hemisphere volume.
$Vhemisphere=12×43π×43=1283πV_{\text{hemisphere}}=\frac{1}{2}\times \frac{4}{3}\pi \times 4^3=\frac{128}{3}\pi$
Add the two volumes.
$Vtotal=48π+1283π=2723πV_{\text{total}}=48\pi+\frac{128}{3}\pi=\frac{272}{3}\pi$
The total volume is $2723π\frac{272}{3}\pi$ cm³.

Tip

In terms of pi

If the question asks for an answer in terms of $π\pi$ , leave $π\pi$ in your answer instead of converting to a decimal.

Working backwards from volume

Sometimes you know the volume and need to find the radius. A cube root undoes cubing.

Example

Sphere and cube with equal volume

A cube has side length 6 cm. A sphere has the same volume as the cube. Find the radius of the sphere to 3 significant figures.

Find the cube volume.
$Vcube=63=216V_{\text{cube}}=6^3=216$
Set the sphere volume equal to 216.
$43πr3=216\frac{4}{3}\pi r^3=216$
Rearrange to find $r^3$ .
$r3=216×34π=162πr^3=\frac{216\times 3}{4\pi}=\frac{162}{\pi}$
Cube root to find $r$ .
$r=162π3≈3.72r=\sqrt[3]{\frac{162}{\pi}}\approx 3.72$
The radius is 3.72 cm.

Water level rise

When a solid sinks fully under water, it pushes up the water by its own volume.

base\text{rise}=\frac{\text{volume of solid}}{\text{area of container base}}

Example

Rise in water level

A rectangular container has base 12 cm by 10 cm. A metal sphere of radius 3 cm sinks fully under the water. Find the rise in water level to 3 significant figures.

A fully submerged sphere displaces its own volume, causing the water level in the rectangular container to rise.

Find the area of the base of the container.
$Abase=12×10=120A_{\text{base}}=12\times 10=120$
Find the volume of the sphere.
$V=43π×33=36πV=\frac{4}{3}\pi \times 3^3=36\pi$
Divide by the base area.
$rise=36π120≈0.942\text{rise}=\frac{36\pi}{120}\approx 0.942$
The water rises by 0.942 cm.

Exam technique

In the exam

First mark the radius; if you see a diameter, halve it.
Check whether the question wants volume, curved surface area, or total surface area.
If a cone surface area question gives vertical height, use Pythagoras to find the slant height.
Leave answers in terms of $π\pi$ when asked; otherwise round only at the final step.

Self review

Check yourself

What extra part must you include for the total surface area of a solid hemisphere?
Which cone formula uses the slant height $l$ ?
How do you turn a displaced volume into a rise in water level?

Spheres and Cones

What you'll learn

Key measurements in diagrams

Volume: space inside

Surface area: outside area

Finding the slant height

Combined shapes

Working backwards from volume

Water level rise

Recap questions

You've reached the end

Spheres and Cones

What you'll learn

Key measurements in diagrams

Volume: space inside

Surface area: outside area

Finding the slant height

Combined shapes

Working backwards from volume

Water level rise

Recap questions

You've reached the end

Spheres and Cones

What you'll learn

Key measurements in diagrams

Volume: space inside

Surface area: outside area

Finding the slant height

Combined shapes

Working backwards from volume

Water level rise

Recap questions

You've reached the end

Spheres and Cones Revision Guide

Spheres and Cones

What you'll learn

Key measurements in diagrams

Volume: space inside

Surface area: outside area

Finding the slant height

Combined shapes

Working backwards from volume

Water level rise

Recap questions

You've reached the end

Spheres and Cones Revision Guide