Area of a Regular n-gon (Radius r)

Statement

The area of a regular polygon (n-gon) can also be expressed using the radius of its circumscribed circle. If each vertex lies on a circle of radius \(R\), then the formula for the area is:

\[ A = \tfrac{n}{2} R^2 \sin\!\left(\tfrac{2\pi}{n}\right) \]

Here, \(n\) is the number of sides, and \(R\) is the radius of the circumscribed circle.

Why it’s true (short reason)

• The polygon can be split into \(n\) identical isosceles triangles, each with central angle \( \tfrac{2\pi}{n} \).
• The area of one triangle is \( \tfrac{1}{2} R^2 \sin(\tfrac{2\pi}{n}) \) (using the formula \( \tfrac{1}{2} ab \sin C \)).
• Multiplying by \(n\) gives the total area of the polygon.

Recipe (how to use it)

1. Identify the number of sides, \(n\), of the regular polygon.
2. Find the radius \(R\) of the circumscribed circle (distance from centre to a vertex).
3. Substitute into the formula \[ A = \tfrac{n}{2} R^2 \sin\!\left(\tfrac{2\pi}{n}\right). \]
4. Evaluate using a calculator, ensuring your calculator is in radians mode if using \(\pi\).

Spotting it

You should use this formula when:

• The problem provides the radius from the centre to a vertex.
• The polygon is explicitly stated as regular.
• The question asks for area but no apothem or side length is given directly.

Common pairings

• Trigonometry, especially the formula for area of a triangle using sine.
• Circle theorems, since the polygon fits perfectly into a circle.
• Comparisons with the apothem-based formula \(A = \tfrac{1}{2} aP\).

Mini examples

1. Given: A regular hexagon with circumradius \(R = 10\). Find: Area. Answer: \(A = \tfrac{6}{2} \times 10^2 \times \sin(60^\circ) = 259.8\ \text{units}^2\).
2. Given: A regular square with radius \(R = 5\). Find: Area. Answer: \(A = \tfrac{4}{2} \times 25 \times \sin(90^\circ) = 50\ \text{units}^2\).

Pitfalls

• Using degree mode instead of radians (be consistent on your calculator).
• Forgetting the factor of \( \tfrac{n}{2} \) in the formula.
• Confusing radius of circumcircle with apothem — they are not the same unless the polygon is a square.
• Rounding too early; carry at least 3–4 significant figures until the final step.

Exam strategy

• Check units — the final answer must be in square units.
• If given side length instead of radius, convert first using trigonometry.
• Remember \(\sin(\tfrac{2\pi}{n})\) is often a known value for common polygons (e.g. \(\sin(60^\circ) = \tfrac{\sqrt{3}}{2}\)).
• Write the formula before substituting numbers to reduce mistakes.

Summary

The radius-based formula \(A = \tfrac{n}{2} R^2 \sin(\tfrac{2\pi}{n})\) arises from splitting the polygon into \(n\) isosceles triangles and applying the sine rule for triangle areas. It is especially useful when the circumradius is given directly. Recognise its use when side length or apothem are absent but the circle radius is known. Accuracy comes from careful substitution and correct use of radians.