Area of a Regular n-gon (Apothem a)

Statement

The area of a regular polygon (an n-gon with all sides and angles equal) can be calculated using the apothem. The apothem is the line from the centre of the polygon, drawn perpendicular to one of its sides. The formula for the area is:

\[ A = \tfrac{1}{2} aP \]

Here, \(A\) is the area, \(a\) is the apothem, and \(P\) is the perimeter (the total length of all sides).

Why it’s true (short reason)

• The polygon can be split into \(n\) congruent isosceles triangles, each with height equal to the apothem.
• The area of each triangle is \(\tfrac{1}{2} \times \text{base} \times \text{height}\).
• Adding all \(n\) triangles gives the total area \(\tfrac{1}{2} aP\).

Recipe (how to use it)

1. Find the perimeter of the polygon: \(P = n \times s\), where \(s\) is the side length.
2. Measure or calculate the apothem \(a\). For a regular polygon this can be done using trigonometry: \[ a = \frac{s}{2\tan(\pi/n)}. \]
3. Substitute into \[ A = \tfrac{1}{2} aP. \]
4. Simplify your result and include correct units (e.g., cm², m²).

Spotting it

You are expected to use this formula when:

• The polygon is regular (all sides and angles equal).
• The apothem is given directly, or can be worked out from geometry or trigonometry.
• The question mentions “area of polygon” together with information about side lengths and perpendicular distance to the centre.

Common pairings

• Perimeter calculations: \(P = ns\).
• Trigonometry in isosceles triangles for finding the apothem.
• Circle geometry, since the apothem is related to the radius of the circumscribed circle.

Mini examples

1. Given: A regular hexagon with side \(10\) cm and apothem \(8.7\) cm. Find: Area. Answer: \(A = \tfrac{1}{2} \times 8.7 \times 60 = 261\ \text{cm}^2\).
2. Given: A square with side \(12\) m. Find: Area using apothem. Answer: Apothem = \(6\), perimeter = \(48\), so \(A = \tfrac{1}{2} \times 6 \times 48 = 144\ \text{m}^2\).

Pitfalls

• Forgetting to check that the polygon is regular. The formula only works for regular polygons.
• Confusing the apothem with the radius (they are different unless the polygon is a square).
• Leaving the area without units. Always include squared units.
• Using the side length instead of the perimeter in the formula.

Exam strategy

• Underline “regular” and “apothem” in the question to avoid misreading.
• Write down \(A = \tfrac{1}{2} aP\) before substituting numbers.
• Check you are multiplying by the whole perimeter, not just one side.
• If only side length is given, compute perimeter first, then apothem (using trigonometry if needed).

Summary

The formula \(A = \tfrac{1}{2} aP\) provides a reliable way to find the area of any regular polygon once the apothem and perimeter are known. It comes from splitting the shape into congruent triangles, each with base on a side and height equal to the apothem. In exams, look for keywords such as “regular polygon” and “apothem” to recognise when to apply it. Always remember to calculate or check the perimeter first, then substitute carefully into the formula.