Circles

Visual model

Radius, diameter and circumference

diameter crosses centreradiuscircumference

C=\pi d\quad\text{and}\quad A=\pi r^{2}

Radius goes from centre to edge.

Diameter is twice the radius.

Keep exact answers in terms of pi when asked.

Gold-standard guide

20 mins

What you will learn

Use radius, diameter, circumference and area.

Use a clear step-by-step method for circles.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier geometry and measures skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Circumference = 2* $\pi$ *r = $\pi$ *d

Step 1

State the radius

r = 9 cm

Step 2

Calculate circumference

C = 2 × $\pi$ × 9 = $18pi c$ m ≈ 56.5 cm (3 s.f.)

Step 3

Calculate area

A = $\pi$ × 9² = $81pi c$ m² ≈ 254 cm² (3 s.f.)

Watch out

Students use radius in the circumference formula where diameter is needed or vice versa

Circumference

$C = 2pi r = \pi d.$

Area

$A = \pi r^{2}.$

Worked example

Find the circumference and area of a circle with radius 9 cm. Give your answers in terms of $\pi a$ nd also to 3 significant figures.

State the radius: r = 9 cm.

Calculate circumference: C = 2 × $\pi$ × 9 = $18pi c$ m ≈ 56.5 cm (3 s.f.).

Calculate area: A = $\pi$ × 9² = $81pi c$ m² ≈ 254 cm² (3 s.f.).

Final answer

Circumference = $18pi$ ≈ 56.5 cm; Area = $81pi$ ≈ 254 cm²

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Find the circumference and area of a circle with radius 9 cm. Give your answers in terms of $\pi a$ nd also to 3 significant figures.

3 marks4 minscircles-worked

Show solution

Worked solution

1.State the radius: r = 9 cm.
2.Calculate circumference: C = 2 × $\pi$ × 9 = $18pi c$ m ≈ 56.5 cm (3 s.f.).
3.Calculate area: A = $\pi$ × 9² = $81pi c$ m² ≈ 254 cm² (3 s.f.).

Final answer

Circumference = $18pi$ ≈ 56.5 cm; Area = $81pi$ ≈ 254 cm²

Mark points

M1: state the radius
M1: calculate circumference
M1: calculate area
A1: Circumference = $18pi$ ≈ 56.5 cm; Area = $81pi$ ≈ 254 cm²

Watch out

Students use radius in the circumference formula where diameter is needed or vice versa.C = $2pi$ *r and C = $\pi$ *d are the same formula. Write both forms and choose based on what is given.

Diagnosticrecall

Find the circumference of a circle with diameter 14 cm. Give in terms of $\pi$ .

1 mark2 minscircles-q1

Show solution

Worked solution

1.Spot the skill: Circumference = 2* $\pi$ *r = $\pi$ *d.
2.Use the state the radius stage first, then calculate circumference.
3.Keep the final answer visible: $14pi c$ m.

Final answer

$14pi c$ m

Mark points

M1: use the correct circumference = 2* $\pi$ *r = $\pi$ *d. area = $\pi$ *r². leave in terms of $\pi f$ or exact answers.memorise both formulae — they are not always given in exams.
A1: $14pi c$ m

Watch out

Students use radius in the circumference formula where diameter is needed or vice versa.C = $2pi$ *r and C = $\pi$ *d are the same formula. Write both forms and choose based on what is given.

Easyprocedure

Find the area of a circle with radius 5 cm, to 1 decimal place.

2 marks3 minscircles-q2

Show solution

Worked solution

1.Spot the skill: Circumference = 2* $\pi$ *r = $\pi$ *d.
2.Use the calculate circumference stage first, then calculate area.
3.Keep the final answer visible: 78.5 cm².

Final answer

78.5 cm²

Mark points

M1: use the correct circumference = 2* $\pi$ *r = $\pi$ *d. area = $\pi$ *r². leave in terms of $\pi f$ or exact answers.memorise both formulae — they are not always given in exams.
A1: 78.5 cm²

Watch out

Students use radius in the circumference formula where diameter is needed or vice versa.C = $2pi$ *r and C = $\pi$ *d are the same formula. Write both forms and choose based on what is given.

Mediumreasoning

A circle has circumference $30pi c$ m. Find its radius and area.

3 marks4 minscircles-q3

Show solution

Worked solution

1.Spot the skill: Circumference = 2* $\pi$ *r = $\pi$ *d.
2.Use the calculate area stage first, then state the radius.
3.Keep the final answer visible: r = 15 cm, area = $225pi c$ m².

Final answer

r = 15 cm, area = $225pi c$ m²

Mark points

M1: use the correct circumference = 2* $\pi$ *r = $\pi$ *d. area = $\pi$ *r². leave in terms of $\pi f$ or exact answers.memorise both formulae — they are not always given in exams.
A1: r = 15 cm, area = $225pi c$ m²

Watch out

Students use radius in the circumference formula where diameter is needed or vice versa.C = $2pi$ *r and C = $\pi$ *d are the same formula. Write both forms and choose based on what is given.

Hardproblem solving

Find the diameter of a circle with area $100pi c$ m².

3 marks5 minscircles-q4

Show solution

Worked solution

1.Spot the skill: Circumference = 2* $\pi$ *r = $\pi$ *d.
2.Use the state the radius stage first, then calculate circumference.
3.Keep the final answer visible: 20 cm.

Final answer

20 cm

Mark points

M1: use the correct circumference = 2* $\pi$ *r = $\pi$ *d. area = $\pi$ *r². leave in terms of $\pi f$ or exact answers.memorise both formulae — they are not always given in exams.
A1: 20 cm

Watch out

Students use radius in the circumference formula where diameter is needed or vice versa.C = $2pi$ *r and C = $\pi$ *d are the same formula. Write both forms and choose based on what is given.

Exam-stylemulti-step

Two circles have radii 3 cm and 5 cm. Find the ratio of their areas.

4 marks6 minscircles-q5

Show solution

Worked solution

1.Spot the skill: Circumference = 2* $\pi$ *r = $\pi$ *d.
2.Use the calculate circumference stage first, then calculate area.
3.Keep the final answer visible: 9:25.

Final answer

9:25

Mark points

M1: use the correct circumference = 2* $\pi$ *r = $\pi$ *d. area = $\pi$ *r². leave in terms of $\pi f$ or exact answers.memorise both formulae — they are not always given in exams.
A1: 9:25

Watch out

Students use radius in the circumference formula where diameter is needed or vice versa.C = $2pi$ *r and C = $\pi$ *d are the same formula. Write both forms and choose based on what is given.

Grade 9 stretchproblem solving

A circular path has outer radius 7 m and inner radius 3 m. Find its area in terms of $\pi$ .

4 marks7 minscircles-g9

Show solution

Worked solution

1.Find both circle areas.
2.Subtract the inner area from the outer area.

Final answer

$40pi m^{2}$

Mark points

M1: use $\pi$ (7² - 3²)
A1: $40pi m^{2}$

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Circles - 2 marksFind the circumference of a circle with diameter 14 cm. Give in terms of

\pi

.Mark answer

Answer

$14pi c$ m

2Angles, lines and polygons - 2 marksThe exterior angle of a regular polygon is 24°. How many sides?Mark answer

Answer

3Properties of shapes - 2 marksName all 2D shapes with equal diagonals that bisect each other.Mark answer

Answer

Rectangle, square

4Perimeter, area and volume - 3 marksFind the perimeter of a rectangle with length 13 cm and width 8 cm.Mark answer

Answer

42 cm

Mastery check

I can explain the method for circles.
I can show clear working without skipping key steps.
I can avoid this mistake: Students use radius in the circumference formula where diameter is needed or vice versa.C = $2pi$ *r and C = $\pi$ *d are the same formula. Write both forms and choose based on what is given.

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Radius, diameter and circumference

What you will learn

Know the rule, then use it

Method

State the radius

Calculate circumference

Calculate area

Watch out

Find the circumference and area of a circle with radius 9 cm. Give your answers in terms of πa\pi aπand also to 3 significant figures.

Build up to the hardest questions

Switch between skills

Get help with anything that still feels tricky.

Find the circumference and area of a circle with radius 9 cm. Give your answers in terms of $\pi a$ nd also to 3 significant figures.