Area of a Circle

Statement

The area of a circle is given by the formula:

\[ A = \pi r^2 \]

Here, \(A\) is the area, \(\pi\) is the mathematical constant approximately equal to 3.1416, and \(r\) is the radius of the circle.

Why it’s true (short reason)

• The circumference of a circle is \(2\pi r\).
• If we “unroll” or “dissect” a circle into triangular slices, they can be rearranged into a shape resembling a rectangle with base \(\pi r\) and height \(r\).
• The area of this rectangle is base × height = \(\pi r \times r = \pi r^2\).

Recipe (how to use it)

1. Identify the radius (half of the diameter).
2. Square the radius: \(r^2\).
3. Multiply by \(\pi\).
4. Attach the correct units (e.g., cm², m²).

Spotting it

• Problems asking for the “area of a circle”.
• Shaded regions involving full circles or circle sectors.
• Composite shapes that include a circle or semicircle.

Common pairings

• Circumference formula: often appears alongside area in geometry questions.
• Sector area: uses the same idea but with a fraction of the circle.
• Density and real-life applications: area of circular bases used in volume calculations.

Mini examples

1. Given: r = 7 cm. Find: area. Answer: \(\pi \times 7^2 = 49\pi\).
2. Given: diameter = 10 cm. Find: area. Answer: radius = 5, so \(25\pi\).

Pitfalls

• Forgetting to halve the diameter: if diameter is given, divide by 2 to find r.
• Forgetting to square r: always compute \(r^2\).
• Mixing circumference with area: circumference is \(2\pi r\), not \(\pi r^2\).

Exam strategy

• Underline whether you’re asked for area or circumference.
• Write down the formula before substituting numbers.
• Check units: if r is in cm, area is in cm².
• Give answers in terms of \(\pi\) unless decimals are requested.

Summary

The area of a circle is \(\pi r^2\). Always square the radius, multiply by \(\pi\), and express the result in square units. This formula is one of the most widely used in GCSE geometry, linking to topics such as sectors, composite areas, and applied problems.