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§ Geometry

Circles

§ Geometry

Circles

CCSS.7.GCCSS.7.G.43 min read

A circle is defined as the set of all points in a plane that are equidistant from a fixed central point. The distance from the centre to any point on the circle is called the radius, whilst the distance across the circle through its centre is the diameter. Key measurements include circumference (perimeter) calculated as 2πr and area calculated as πr².

§ 01

Why it matters

Circles appear throughout engineering, architecture, and daily life. Car wheels, clock faces, and sports fields all rely on circular geometry. In construction, architects use circle calculations to design domes and arches — the London Eye's circumference of approximately 424 metres required precise circular mathematics. Pizza companies calculate areas to price different sizes fairly: a 12-inch pizza has roughly 113 square inches compared to 79 square inches for a 10-inch pizza. Circle theorems become essential in GCSE mathematics, particularly for Year 10 and 11 students studying angles in circles, tangent properties, and chord relationships. Advanced applications include satellite orbits, wheel mechanics, and optical lens design, where understanding circular properties determines functionality and efficiency.

§ 02

How to solve circles

Circles — Circumference & Area

  • Circumference = 2πr (or πd).
  • Area = πr².
  • Use π ≈ 3.14 unless told otherwise.
  • Diameter = 2 × radius.

Example: r = 5: C = 2π(5) = 31.4, A = π(25) ≈ 78.5.

§ 03

Worked examples

Beginner§ 01

The radius of a circle is 2 cm. What is the diameter?

Answer: 4 cm

  1. Diameter = 2 × radius → 2 × 2 = 4 cm — The diameter is always twice the radius.
Easy§ 02

Find the circumference of a circle with radius 11 cm (use π ≈ 3.14).

Answer: ≈ 69.12 cm

  1. Apply formula: C = 2πr → C = 2 × π × 11 ≈ 69.12 cm — Circumference = 2 × π × 11 ≈ 69.12 cm.
Medium§ 03

Find the area of a circle with radius 9 cm.

Answer: ≈ 254.47 cm²

  1. Apply formula: A = πr² → A = π × 9² = π × 81 ≈ 254.47 cm² — Area = π × 9² = π × 81 ≈ 254.47 cm².
§ 04

Common mistakes

  • Confusing radius and diameter leads to errors like calculating circumference as 2π(10) = 62.8 cm when the diameter is 10 cm, instead of using radius 5 cm to get 31.4 cm
  • Forgetting to square the radius in area calculations produces A = π × 6 = 18.8 cm² instead of A = π × 6² = 113.1 cm² for a circle with radius 6 cm
  • Using the wrong formula entirely, such as calculating area as 2πr instead of πr², giving 37.7 cm² rather than the correct 113.1 cm² for radius 6 cm
§ 05

Frequently asked questions

What is the difference between radius and diameter?
Radius is the distance from centre to edge, whilst diameter is the distance across the entire circle through the centre. Diameter always equals twice the radius, so a circle with radius 8 cm has diameter 16 cm.
Why do we use π in circle calculations?
π represents the constant ratio between a circle's circumference and its diameter, approximately 3.14159. This ratio remains the same for all circles regardless of size, making π essential for accurate circumference and area calculations.
How do you find radius from area?
Rearrange the area formula A = πr² to get r = √(A/π). For example, if area is 78.54 cm², then r = √(78.54/π) = √25 = 5 cm. This reverse calculation is useful in real-world applications.
When should I use 3.14 versus the π button on my calculator?
Use π ≈ 3.14 for simple approximations or when specifically instructed. Use the π button for more precise calculations, especially in GCSE exams where accuracy matters. The π button gives more decimal places for better precision.
What are the main circle theorems in GCSE?
Key theorems include: angles in semicircles are 90°, angles at the centre are twice angles at the circumference, and tangents from external points are equal. These appear prominently in Year 10-11 geometry.
§ 06

See also

SetRadiusDiameterCircumferencePerimeterAreaRatio
§ 06

Where to next?

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