Area and Circumference of Circles

What you'll learn

How to name the key parts of a circle, including radius, diameter, tangent and sector.
How to use the formulas for circumference and area.
How to leave answers in terms of $π\pi$ or round to a given number of decimal places.
How to solve problems with semicircles, sectors, rings and composite shapes.

1. Circle vocabulary

A circle is a perfectly round shape where every point on the edge is the same distance from the centre.

This diagram shows the circle words you need to recognise before using the formulas.

Labelled diagram showing radius, diameter, circumference, sector and tangent of a circle

Definition

Key circle words

The centre is the exact middle of the circle.
A radius is a straight line from the centre to the edge.
A diameter is a straight line across the circle through the centre.
The circumference is the distance all the way around the circle.
A tangent is a straight line that touches the circle at exactly one point.
A sector is a “slice” of a circle, made from two radii and an arc.

Example

Naming parts of a circle

If a line goes from the centre to the edge, its mathematical name is radius.
If a line goes across the whole circle through the centre, its mathematical name is diameter.
If a line touches the circle at one point only, its mathematical name is tangent.
If a shaded part looks like a pizza slice, its mathematical name is sector.

2. Radius and diameter

The diameter is always twice the radius.

d = 2 r

The radius is always half the diameter.

\frac{d}{2}

Key Idea

Radius and diameter

Most circle formula mistakes happen before the formula is used. Always check whether the question gives you the radius or the diameter.

Example

Finding the radius from the diameter

A circle has diameter 9 m.
The radius is half the diameter.
$\frac{9}{2} = 4.5$
The radius is 4.5 m.

3. Circumference of a circle

The circumference is the perimeter of a circle: the distance around the outside.

The two circumference formulas are:

2\pi r

and

\pi d

Use $2\pi r$ when you know the radius. Use $\pi d$ when you know the diameter.

Example

Circumference from the radius

A circle has radius 6.5 cm, so use $2\pi r$ .
Substitute the radius into the formula.
$2\pi \times 6.5 = 13\pi$
Use the calculator value of $π\pi$ .
$13π≈40.840713\pi \approx 40.8407$
The circumference is 40.84 cm, correct to 2 decimal places.

Common Mistake

Using the diameter formula with a radius

If the question gives the radius, do not use $\pi d$ unless you first double the radius to find the diameter.

Real-life circumference costs

If something goes around the edge of a circular object, you usually need the circumference.

Example

Cost of fencing a circular field

A circular field has diameter 28 m, so use $\pi d$ .
Find the distance around the field.
$\pi \times 28 \approx 87.965$
If fencing costs £14.50 per metre, multiply the circumference by 14.50.
$87.965 \times 14.50 \approx 1275.49$
The total cost is £1275.49.

4. Area of a circle

The area of a circle is the space inside it.

The area formula is:

\pi r^2

This means “pi times the radius squared”.

Example

Area from the diameter

A circle has diameter 9 m, so first find the radius.
$\frac{9}{2} = 4.5$
Use the area formula.
$\pi r^2$
Substitute $r = 4.5$ .
$\pi \times 4.5^2 = 20.25\pi$
Use a calculator.
$20.25π≈63.61720.25\pi \approx 63.617$
The area is 63.6 m², correct to 1 decimal place.

Tip

Squaring the radius

For area, square the radius, not the diameter. On a calculator, you can type $π×r×r\pi \times r \times r$ .

5. Answers in terms of $π\pi$

Sometimes the question asks for an answer in terms of $π\pi$ . This means you should leave $π\pi$ in the answer and not turn it into a decimal.

Example

Leaving answers in terms of pi

A circle has diameter 12 mm. For circumference, use $\pi d$ .
$\pi \times 12 = 12\pi$
So the exact circumference is $12π12\pi$ mm.
A circle has radius 8 cm. For area, use $\pi r^2$ .
$\pi \times 8^2 = 64\pi$
So the exact area is $64π64\pi$ cm².

Tip

When to round

If the question says “in terms of $π\pi$ ”, do not round. If it says “correct to 1 decimal place” or “correct to 2 decimal places”, use a decimal answer.

6. Semicircles and sectors

A semicircle is half a circle. Its curved edge is half of the full circumference, but its perimeter also includes the straight diameter.

A sector is a fraction of a circle. For example, a 90° sector is one quarter of a circle, and a 270° sector is three quarters of a circle.

These three shapes appear often in exam questions.

Diagrams of a semicircle, an annulus ring and a three-quarter circle sector

Common Mistake

Perimeter of a semicircle

The perimeter of a semicircle is not just half the circumference. You must also add the diameter.

Example

Finding the perimeter of a semicircle from its area

A semicircle has area 50 m². Therefore the full circle would have area 100 m².
Use $\pi r^2$ for the full circle.
$πr2=100\pi r^2 = 100$
Rearrange to find the radius.
$r2=100πr^2 = \frac{100}{\pi}$
Square root to find the radius.
$\approx 5.642$
A semicircle perimeter is curved edge plus diameter.
$\pi r + 2r$
Substitute the radius.
$\approx \pi(5.642) + 2(5.642) \approx 29.008$
The perimeter is 29.0 m, correct to 1 decimal place.

Example

Perimeter of a three-quarter circle

The shape has radius 10 m. The curved part is three quarters of the full circumference.
$34×2π×10=15π\frac{3}{4} \times 2\pi \times 10 = 15\pi$
Add the two straight radii.
$15\pi + 10 + 10$
Calculate the total perimeter.
$15π+20≈67.12415\pi + 20 \approx 67.124$
The perimeter is 67.1 m, correct to 1 decimal place.

7. Composite circle problems

A composite shape is made by joining or cutting out simpler shapes. The main strategy is:

Add areas when shapes are joined together.
Subtract areas when a shape is cut out or left unshaded.
For perimeters, only include the outside edges.

Example

Area of a ring

A ring is made by cutting a smaller circle of radius 7 cm out of a larger circle with diameter 18 cm.
The larger radius is half of 18 cm.
$R = 9$
Find the two circle areas.
$Alarge=π×92=81πA_{\text{large}} = \pi \times 9^2 = 81\pi$
Find the smaller circle area.
$Asmall=π×72=49πA_{\text{small}} = \pi \times 7^2 = 49\pi$
Subtract to find the shaded ring.
$Aring=81π−49π=32πA_{\text{ring}} = 81\pi - 49\pi = 32\pi$
The area of the ring is about 100.5 cm².

Example

How many boxes are needed?

A garden is made from a square and a semicircle. The semicircle has radius 5 m, so its diameter is 10 m.
The square has side length 10 m.
$area=10×10=100\text{square area} = 10 \times 10 = 100$
Find the semicircle area.
$area=12π×52≈39.27\text{semicircle area} = \frac{1}{2}\pi \times 5^2 \approx 39.27$
Add the two areas.
$area≈100+39.27=139.27\text{total area} \approx 100 + 39.27 = 139.27$
If one box covers 30 m², divide the total area by 30.
$139.27 \div 30 \approx 4.64$
You need 5 boxes, because 4 boxes would not cover the whole garden.

Exam technique

In the exam

Start by labelling the radius and diameter on the diagram before choosing a formula.
Use the calculator $π\pi$ button, and keep full calculator values until the final rounding step.
For shaded regions, write a clear “big area minus small area” or “part plus part” calculation.

Self review

Check yourself

If a circle has diameter 14 cm, what is its radius?
What is the difference between circumference and area?
When finding the perimeter of a semicircle, which straight edge must you remember to add?

Area and Circumference of Circles

What you'll learn

How to name the key parts of a circle, including radius, diameter, tangent and sector.
How to use the formulas for circumference and area.
How to leave answers in terms of $π\pi$ or round to a given number of decimal places.
How to solve problems with semicircles, sectors, rings and composite shapes.

1. Circle vocabulary

A circle is a perfectly round shape where every point on the edge is the same distance from the centre.

This diagram shows the circle words you need to recognise before using the formulas.

Labelled diagram showing radius, diameter, circumference, sector and tangent of a circle

Definition

Key circle words

The centre is the exact middle of the circle.
A radius is a straight line from the centre to the edge.
A diameter is a straight line across the circle through the centre.
The circumference is the distance all the way around the circle.
A tangent is a straight line that touches the circle at exactly one point.
A sector is a “slice” of a circle, made from two radii and an arc.

Example

Naming parts of a circle

If a line goes from the centre to the edge, its mathematical name is radius.
If a line goes across the whole circle through the centre, its mathematical name is diameter.
If a line touches the circle at one point only, its mathematical name is tangent.
If a shaded part looks like a pizza slice, its mathematical name is sector.

2. Radius and diameter

The diameter is always twice the radius.

d = 2 r

The radius is always half the diameter.

\frac{d}{2}

Key Idea

Radius and diameter

Most circle formula mistakes happen before the formula is used. Always check whether the question gives you the radius or the diameter.

Example

Finding the radius from the diameter

A circle has diameter 9 m.
The radius is half the diameter.
$\frac{9}{2} = 4.5$
The radius is 4.5 m.

3. Circumference of a circle

The circumference is the perimeter of a circle: the distance around the outside.

The two circumference formulas are:

2\pi r

and

\pi d

Use $2\pi r$ when you know the radius. Use $\pi d$ when you know the diameter.

Example

Circumference from the radius

A circle has radius 6.5 cm, so use $2\pi r$ .
Substitute the radius into the formula.
$2\pi \times 6.5 = 13\pi$
Use the calculator value of $π\pi$ .
$13π≈40.840713\pi \approx 40.8407$
The circumference is 40.84 cm, correct to 2 decimal places.

Common Mistake

Using the diameter formula with a radius

If the question gives the radius, do not use $\pi d$ unless you first double the radius to find the diameter.

Real-life circumference costs

If something goes around the edge of a circular object, you usually need the circumference.

Example

Cost of fencing a circular field

A circular field has diameter 28 m, so use $\pi d$ .
Find the distance around the field.
$\pi \times 28 \approx 87.965$
If fencing costs £14.50 per metre, multiply the circumference by 14.50.
$87.965 \times 14.50 \approx 1275.49$
The total cost is £1275.49.

4. Area of a circle

The area of a circle is the space inside it.

The area formula is:

\pi r^2

This means “pi times the radius squared”.

Example

Area from the diameter

A circle has diameter 9 m, so first find the radius.
$\frac{9}{2} = 4.5$
Use the area formula.
$\pi r^2$
Substitute $r = 4.5$ .
$\pi \times 4.5^2 = 20.25\pi$
Use a calculator.
$20.25π≈63.61720.25\pi \approx 63.617$
The area is 63.6 m², correct to 1 decimal place.

Tip

Squaring the radius

For area, square the radius, not the diameter. On a calculator, you can type $π×r×r\pi \times r \times r$ .

5. Answers in terms of $π\pi$

Sometimes the question asks for an answer in terms of $π\pi$ . This means you should leave $π\pi$ in the answer and not turn it into a decimal.

Example

Leaving answers in terms of pi

A circle has diameter 12 mm. For circumference, use $\pi d$ .
$\pi \times 12 = 12\pi$
So the exact circumference is $12π12\pi$ mm.
A circle has radius 8 cm. For area, use $\pi r^2$ .
$\pi \times 8^2 = 64\pi$
So the exact area is $64π64\pi$ cm².

Tip

When to round

If the question says “in terms of $π\pi$ ”, do not round. If it says “correct to 1 decimal place” or “correct to 2 decimal places”, use a decimal answer.

6. Semicircles and sectors

A semicircle is half a circle. Its curved edge is half of the full circumference, but its perimeter also includes the straight diameter.

A sector is a fraction of a circle. For example, a 90° sector is one quarter of a circle, and a 270° sector is three quarters of a circle.

These three shapes appear often in exam questions.

Diagrams of a semicircle, an annulus ring and a three-quarter circle sector

Common Mistake

Perimeter of a semicircle

The perimeter of a semicircle is not just half the circumference. You must also add the diameter.

Example

Finding the perimeter of a semicircle from its area

A semicircle has area 50 m². Therefore the full circle would have area 100 m².
Use $\pi r^2$ for the full circle.
$πr2=100\pi r^2 = 100$
Rearrange to find the radius.
$r2=100πr^2 = \frac{100}{\pi}$
Square root to find the radius.
$\approx 5.642$
A semicircle perimeter is curved edge plus diameter.
$\pi r + 2r$
Substitute the radius.
$\approx \pi(5.642) + 2(5.642) \approx 29.008$
The perimeter is 29.0 m, correct to 1 decimal place.

Example

Perimeter of a three-quarter circle

The shape has radius 10 m. The curved part is three quarters of the full circumference.
$34×2π×10=15π\frac{3}{4} \times 2\pi \times 10 = 15\pi$
Add the two straight radii.
$15\pi + 10 + 10$
Calculate the total perimeter.
$15π+20≈67.12415\pi + 20 \approx 67.124$
The perimeter is 67.1 m, correct to 1 decimal place.

7. Composite circle problems

A composite shape is made by joining or cutting out simpler shapes. The main strategy is:

Add areas when shapes are joined together.
Subtract areas when a shape is cut out or left unshaded.
For perimeters, only include the outside edges.

Example

Area of a ring

A ring is made by cutting a smaller circle of radius 7 cm out of a larger circle with diameter 18 cm.
The larger radius is half of 18 cm.
$R = 9$
Find the two circle areas.
$Alarge=π×92=81πA_{\text{large}} = \pi \times 9^2 = 81\pi$
Find the smaller circle area.
$Asmall=π×72=49πA_{\text{small}} = \pi \times 7^2 = 49\pi$
Subtract to find the shaded ring.
$Aring=81π−49π=32πA_{\text{ring}} = 81\pi - 49\pi = 32\pi$
The area of the ring is about 100.5 cm².

Example

How many boxes are needed?

A garden is made from a square and a semicircle. The semicircle has radius 5 m, so its diameter is 10 m.
The square has side length 10 m.
$area=10×10=100\text{square area} = 10 \times 10 = 100$
Find the semicircle area.
$area=12π×52≈39.27\text{semicircle area} = \frac{1}{2}\pi \times 5^2 \approx 39.27$
Add the two areas.
$area≈100+39.27=139.27\text{total area} \approx 100 + 39.27 = 139.27$
If one box covers 30 m², divide the total area by 30.
$139.27 \div 30 \approx 4.64$
You need 5 boxes, because 4 boxes would not cover the whole garden.

Exam technique

In the exam

Start by labelling the radius and diameter on the diagram before choosing a formula.
Use the calculator $π\pi$ button, and keep full calculator values until the final rounding step.
For shaded regions, write a clear “big area minus small area” or “part plus part” calculation.

Self review

Check yourself

If a circle has diameter 14 cm, what is its radius?
What is the difference between circumference and area?
When finding the perimeter of a semicircle, which straight edge must you remember to add?

Area and Circumference of Circles

What you'll learn

1. Circle vocabulary

2. Radius and diameter

3. Circumference of a circle

Real-life circumference costs

4. Area of a circle

5. Answers in terms of $π\pi$

6. Semicircles and sectors

7. Composite circle problems

Recap questions

You've reached the end

Area and Circumference of Circles

What you'll learn

1. Circle vocabulary

2. Radius and diameter

3. Circumference of a circle

Real-life circumference costs

4. Area of a circle

5. Answers in terms of $π\pi$

6. Semicircles and sectors

7. Composite circle problems

Recap questions

You've reached the end

Area and Circumference of Circles

What you'll learn

1. Circle vocabulary

2. Radius and diameter

3. Circumference of a circle

Real-life circumference costs

4. Area of a circle

5. Answers in terms of π\piπ

6. Semicircles and sectors

7. Composite circle problems

Recap questions

You've reached the end

Area and Circumference of Circles Revision Guide

Area and Circumference of Circles

What you'll learn

1. Circle vocabulary

2. Radius and diameter

3. Circumference of a circle

Real-life circumference costs

4. Area of a circle

5. Answers in terms of π\piπ

6. Semicircles and sectors

7. Composite circle problems

Recap questions

You've reached the end

Area and Circumference of Circles Revision Guide

5. Answers in terms of $π\pi$

5. Answers in terms of $π\pi$