Circumference of a Circle

Statement

The circumference of a circle is the distance all the way around its boundary. It can be thought of as the perimeter of the circle. The formula can be written in two ways, depending on whether the radius \(r\) or diameter \(d\) is used:

\[ C = 2\pi r \quad \text{or} \quad C = \pi d \]

Why it’s true

• Circles are perfectly symmetrical, and their perimeter grows in direct proportion to the radius or diameter.
• The number \(\pi\) represents the constant ratio between circumference and diameter: \(\pi = \frac{C}{d}\).
• Multiplying diameter by \(\pi\) gives the circumference. Since diameter is twice the radius, this leads to \(C = 2\pi r\).

Recipe (how to use it)

1. Identify whether the problem gives you the radius or the diameter.
2. If radius is given, use \(C = 2\pi r\).
3. If diameter is given, use \(C = \pi d\).
4. Substitute the value and calculate. Use \(\pi \approx 3.14\) or leave your answer in terms of \(\pi\), depending on the instructions.

Spotting it

Whenever a problem asks for “perimeter of a circle”, “length around”, or “circumference”, you must use this formula. If only radius is given, double it to find the diameter first if needed.

Common pairings

• Area of a circle (\(A = \pi r^2\)).
• Arc length, which is a fraction of circumference.
• Sectors, combining arc length and area together.

Mini examples

1. Given: radius = 7 cm. Find: circumference. Answer: \(C = 2\pi(7) = 14\pi \,\text{cm}\).
2. Given: diameter = 10 cm. Find: circumference. Answer: \(C = 10\pi \,\text{cm}\).

Pitfalls

• Confusing diameter and radius: Always check whether the question provides diameter or radius.
• Forgetting units: Circumference is a length, so always give units such as cm or m.
• Premature rounding: Only round at the final step unless otherwise told.
• Mixing up formulas: Do not confuse circumference (\(2\pi r\)) with area (\(\pi r^2\)).

Exam strategy

• Underline whether the question asks for area or circumference.
• Write down the correct formula before substituting values.
• Decide whether the answer should be exact (with \(\pi\)) or approximate (decimal).

Summary

The circumference of a circle is the distance around its boundary, found by multiplying the diameter by \(\pi\) or the radius by \(2\pi\). This is one of the most fundamental circle formulas and is often used alongside area and arc length in GCSE exams. Always pay attention to whether you are given radius or diameter, and ensure your final answer includes units.