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Circumference from Radius and Diameter Revision Notes | GCSE AQA Higher Maths

The circumference of a circle is C = 2πr when the radius is known, or C = πd when the diameter is known. Free GCSE Maths notes and 7 FAQs.

Updated 25 Mar 2026 · Revise Right Team

Circumference from Radius and Diameter is part of Mensuration and calculation in GCSE AQA Higher Maths.

4 min

Reading time

42

Lessons in this topic

GCSE AQA Higher

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Reviewed by Revise Right

Content review

What this lesson covers

The circumference of a circle is C = 2πr when the radius is known, or C = πd when the diameter is known. Free GCSE Maths notes and 7 FAQs.

  • Recall and apply the formulae for the circumference of a circle: $C = 2\pi r = \pi d$
  • Recall and apply the formula for the area of a circle: $A = \pi r^2$
  • Calculate the perimeter of 2D shapes that include circular arcs or full circles
  • Work with exact answers in terms of $\pi$ and rounded decimal answers

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Key ideas to keep in view

Use this lesson to stay inside Mensuration and calculation while connecting the detail back to the wider Geometry and measures topic.

  • Circumference: The perimeter of a circle — the total distance around its edge. Calculated using $C = \pi d$ or $C = 2\pi r$.
  • Radius: The distance from the centre of a circle to any point on its edge. Always half the diameter.
  • Diameter: The distance across a circle through its centre. Always twice the radius: $d = 2r$.
  • Pi ($\pi$): A mathematical constant approximately equal to 3.14159. It is the ratio of any circle's circumference to its diameter.

Revision notes

Read the core explanations from Circumference from Radius and Diameter before testing yourself from memory.

1. Why Circles Need Their Own Formulae

You can find the perimeter of a rectangle by adding up its four straight sides. However, circles are entirely curved and have no straight edges, so the same approach does not work. Instead, we rely on the constant \(\pi\), which serves as a unique bridge between the linear dimension (the distance around the circle) and the circle's curved nature. In contrast to polygons, where the perimeter is simply the sum of straight sides, circular shapes require a completely different method of measurement. The number \(\pi\) is fundamental because it represents the constant ratio between a circle’s circumference and its diameter. This property not only helps us calculate the circumference but is also essential when determining the area of a circle. In addition to these geometrical benefits, the use of \(\pi\) has historical significance, having been studied since ancient times. Its approximate value of 3.14159 is used universally and appears in numerous mathematical areas such as trigonometry and calculus. A solid understanding of \(\pi\) and the formulas that use it is crucial for solving exam problems and for a deeper appreciation of mathematical principles.

Every circle, regardless of its size, maintains a consistent ratio between its circumference and diameter. This ratio is given by \(\pi \approx 3.14159\), a number that remains identical for all circles. This remarkable and timeless property forms one of the cornerstones of circle geometry. Not only does it allow us to calculate the distance around a circle (its circumference), but it also underpins the formula for the area of a circle. Understanding this invariant ratio is essential because it simplifies complex problems and provides a reliable method to tackle exam questions. Moreover, the appearance of \(\pi\) in so many different formulas reveals the deep connection between linear and circular measurements, making it a fundamental constant across various branches of mathematics. Mastery of these concepts will not only help in achieving success in GCSE examinations but also in further mathematical studies.

2. The Two Circle Formulae You Must Know

You need to know both the circumference formula and the area formula. They look similar but measure completely different things — one is a length, the other is a flat region.

Notice that $C = 2\pi r$ and $C = \pi d$ are the same formula written two ways. Since $d = 2r$, substituting gives $\pi d = \pi \times 2r = 2\pi r$. You choose whichever version matches the measurement you have been given.

3. Calculating Circumference — Worked Examples

Theory is only useful once you can apply it under exam conditions. The two worked examples below cover the two most common question types: given the radius, and given the diameter.

Worked example: Calculate the Circumference Given the Radius

A circular pond has a radius of 6 m. Calculate its circumference. Give your answer to 1 decimal place.

  1. $C = 2\pi r$ — The radius is given, so use $C = 2\pi r$. [M1 — correct formula selected]
  2. $C = 2 \times \pi \times 6$ — Replace $r$ with 6. Keep $\pi$ as a symbol for now — do not round early. [M1 — correct substitution]
  3. $C = 12\pi$ — $2 \times 6 = 12$, so $C = 12\pi$ m. This is the exact answer.
  4. $C = 12 \times 3.14159... = 37.699...$ $C \approx 37.7 \text{ m}$ — Only round at the very last step. [A1 — correct answer with units]

Answer: $C = 12\pi \approx 37.7$ m (to 1 d.p.)

Worked example: Calculate the Circumference Given the Diameter

A wheel has a diameter of 50 cm. Calculate its circumference. Give your answer to 3 significant figures.

  1. $C = \pi d$ — The diameter is given directly, so $C = \pi d$ is the efficient choice. [M1]
  2. $C = \pi \times 50$ — Replace $d$ with 50. [M1 — correct substitution]
  3. $C = 50\pi$ cm — This is the exact answer. Acceptable as a final answer if the question asks for exact form.
  4. $C = 50 \times 3.14159... = 157.079...$ $C \approx 157 \text{ cm}$ — 157.079... rounds to 157 to 3 s.f. [A1 — correct answer with units]

Answer: $C = 50\pi \approx 157$ cm (to 3 s.f.)

4. Calculating Area — Worked Examples

The area formula $A = \pi r^2$ always requires the radius. If a question gives you the diameter, your very first step must be to halve it. Skipping this step is one of the most common errors in circle questions. When calculating with the diameter, remember to convert it to the radius first. This ensures accuracy in all areas requiring circular calculations.

Worked example: Calculate the Area Given the Radius

Calculate the area of a circle with radius 7 cm. Give your answer in terms of $\pi$.

  1. $A = \pi r^2$ — Area of a circle always uses the radius squared. [M1]
  2. $A = \pi \times 7^2$ — Replace $r$ with 7.
  3. $7^2 = 7 \times 7 = 49$ $A = 49\pi$ — $7 \times 7 = 49$. The question asks for the exact answer, so leave in terms of $\pi$. [A1]

Answer: $A = 49\pi$ cm²

Worked example: Calculate the Area Given the Diameter

A circular table top has a diameter of 120 cm. Calculate its area. Give your answer to the nearest cm².

  1. $r = \frac{d}{2} = \frac{120}{2} = 60$ cm — Always halve the diameter before using the area formula. [M1 — correct conversion]
  2. $A = \pi r^2$ — Area formula requires the radius.
  3. $A = \pi \times 60^2$ $60^2 = 60 \times 60 = 3600$ $A = 3600\pi$ — $60 \times 60 = 3600$. Exact form is $3600\pi$ cm².
  4. $A = 3600 \times 3.14159... = 11309.73...$ $A \approx 11310 \text{ cm}^2$ — Round only at the final step. [A1 — correct answer with units]

Answer: $A \approx 11310$ cm² (to the nearest cm²)

5. Perimeters of 2D Shapes Involving Circles

Many exam questions combine a semicircle or quarter-circle with straight edges. The perimeter of such a shape is the total distance around its outside — which means adding the curved arc length to the straight sides.

A semicircle's curved edge is exactly half the full circumference: $\frac{1}{2} \times 2\pi r = \pi r$. A quarter-circle's curved edge is $\frac{1}{4} \times 2\pi r = \frac{\pi r}{2}$. The straight edges (the diameter or radii) must be added separately.

Worked example: Perimeter of a Semicircle

A semicircle has a diameter of 14 cm. Calculate its perimeter. Give your answer to 1 decimal place.

  1. Perimeter = curved arc + straight diameter — The perimeter goes around the curved edge AND along the flat (diameter) edge. Both must be included.
  2. $r = \frac{14}{2} = 7$ cm — Halve the diameter to get the radius for the arc formula. [M1]
  3. Arc $= \frac{1}{2} \times 2\pi r = \pi r = \pi \times 7 = 7\pi$ cm — Half the full circumference. [M1 — correct arc formula]
  4. Perimeter $= 7\pi + 14$ — The straight edge is the full diameter, 14 cm.
  5. $7\pi = 7 \times 3.14159... = 21.991...$ Perimeter $= 21.991... + 14 = 35.991...$ Perimeter $\approx 36.0$ cm — Add the decimal arc to the straight edge, then round. [A1 — correct answer with units]

Answer: Perimeter $= 7\pi + 14 \approx 36.0$ cm (to 1 d.p.)

Worked example: Perimeter of a Composite Shape (Quarter-Circle and Rectangle)

A shape is made from a rectangle 10 cm by 6 cm with a quarter-circle of radius 6 cm removed from one corner. Calculate the perimeter of the remaining shape. Give your answer to 1 decimal place.

  1. Rectangle sides: 10 cm and 6 cm. Quarter-circle radius: 6 cm, cut from one corner. — Visualising the shape prevents missing edges. The two sides adjacent to the cut corner are replaced by the curved arc.
  2. One full long side (10 cm), one full short side (6 cm), one partial long side, one short side completely replaced by the arc, and the quarter-circle arc. — When a quarter-circle is removed from a corner, the side along that corner is partially lost. Here, the remaining lengths are calculated from the original dimensions.
  3. Arc = 1/4 × 2πr = (πr)/2 = (π × 6)/2 = 3π cm — A quarter of the full circumference. [M1]
  4. Top (opposite long side): 10 cm\nRight (opposite short side): 6 cm\nBottom (partial long side): 10 - 6 = 4 cm\nLeft (fully consumed short side): 6 - 6 = 0 cm — The straight edges comprise the untouched long side (10 cm), the opposite short side (6 cm), the remaining part of the long side (4 cm), and a short side that is completely replaced by the arc (0 cm).
  5. Perimeter = 10 + 6 + 4 + 3π = 20 + 3π\n= 20 + 9.4247... ≈ 29.4 cm — Add the straight edge lengths and the arc length, then round appropriately. [A1]

Answer: Perimeter = 20 + 3π ≈ 29.4 cm (to 1 d.p.)

6. Exam Technique and Common Mistakes

Circle questions on GCSE Higher exam papers test both your formula knowledge and your attention to detail. The mistakes below are responsible for a large proportion of lost marks — recognising them now means you will not repeat them under pressure.

Frequently asked questions

What is pi?

π is the constant ratio between a circle's circumference and its diameter, approximately 3.14159, and it is the same for every circle. It appears in both C = πd and A = πr².

What is the formula for the circumference of a circle?

The circumference of a circle is C = 2πr when the radius is known, or C = πd when the diameter is known. A pond with radius 6 m has a circumference of 12π ≈ 37.7 m, and a wheel with diameter 50 cm has a circumference of 50π ≈ 157 cm.

How do you find the perimeter of a semicircle?

Add the curved arc, which is half the circumference (πr), to the straight diameter. For a diameter of 14 cm, the radius is 7 cm, so the perimeter is 7π + 14 ≈ 36.0 cm to 1 decimal place.

Is circumference 2πr or πd?

C = 2πr and C = πd are the same formula written two ways: since d = 2r, πd = π × 2r = 2πr. The version chosen should match the measurement given in the question.

What is the arc length of a quarter circle?

The arc length of a quarter circle is ¼ × 2πr = πr/2. For radius 6 cm, the arc is (π × 6)/2 = 3π cm, and any straight edges are added separately for a perimeter.

How do you find the perimeter of a shape with a quarter-circle?

Add the straight edges and the curved arc, which is a quarter of the circumference. A 10 cm by 6 cm rectangle with a quarter-circle of radius 6 cm removed from a corner has an arc of 3π cm and straight edges of 10 + 6 + 4, so its perimeter is 20 + 3π ≈ 29.4 cm.

What does exact form mean with π?

Exact form means leaving the answer in terms of π, such as 12π cm, rather than converting to a decimal. It is a perfectly precise result and is often what a question asks for.

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