How to Find the Area of a Regular Polygon at GCSE

What is a regular polygon?

A regular polygon is a closed flat shape in which every side has the same length and every interior angle has the same measure. Examples include equilateral triangles (3 sides), squares (4 sides), regular pentagons (5 sides), regular hexagons (6 sides), and so on.

The interior angle of a regular n-sided polygon is given by:

Interior angle = (n − 2) × 180° ÷ n

For a regular hexagon (n = 6): interior angle = 4 × 180° ÷ 6 = 120°.

What is the apothem?

The apothem is the perpendicular distance from the centre of a regular polygon to the midpoint of one of its sides. It is the height of each triangle when you split the polygon into congruent isosceles triangles from the centre.

There are two common methods for finding the area of a regular polygon:

• Method 1 — Splitting into triangles: Divide the polygon into n congruent isosceles triangles, find the area of one, and multiply by n.
• Method 2 — Apothem formula: Use Area = ½ × perimeter × apothem.

Both methods give the same result.

How do you find the area of a regular polygon step by step?

Using the triangle method:

1. Identify n and the side length s. Write down the number of sides and the length of each side.
2. Find the central angle. Each triangle at the centre has an angle of 360° ÷ n.
3. Find the apothem using trigonometry. The apothem is the adjacent side in a right-angled triangle where the angle at the centre is (360° ÷ n) ÷ 2 = 180° ÷ n and the opposite side is s ÷ 2. So: apothem = (s ÷ 2) ÷ tan(180° ÷ n).
4. Find the area of one triangle. Area of one triangle = ½ × base × height = ½ × s × apothem.
5. Multiply by n. Total area = n × (½ × s × apothem) = ½ × n × s × apothem = ½ × perimeter × apothem.

Worked example: area of a regular hexagon

Find the area of a regular hexagon with side length 8 cm.

Step 1 — Values: n = 6, s = 8 cm

Step 2 — Central angle per triangle: 360° ÷ 6 = 60°. Each triangle has a top angle of 60°.

Step 3 — Find the apothem:

Half-angle at centre = 30° Opposite side = s ÷ 2 = 4 cm

apothem = 4 ÷ tan(30°) = 4 ÷ (1/√3) = 4√3 cm ≈ 6.928 cm

Step 4 — Area of one triangle:

Area = ½ × 8 × 4√3 = 16√3 cm²

Step 5 — Total area:

Area = 6 × 16√3 = 96√3 ≈ 166.3 cm²

Using the apothem formula as a check:

Perimeter = 6 × 8 = 48 cm

Area = ½ × 48 × 4√3 = 96√3 ≈ 166.3 cm² ✓

Interior angles, apothems and areas for common regular polygons

The table below gives values for a regular polygon with side length 1 unit. Multiply the area by s² (where s is your actual side length in any unit) to scale to any size.

Example using the table: Area of a regular pentagon with side 5 cm: Area = 1.720 × 5² = 1.720 × 25 = 43.0 cm²

What if you are given the apothem directly?

If the question gives you the apothem rather than the side length, use:

Area = ½ × perimeter × apothem

You will need to find the side length from the apothem. Using the same right-angled triangle:

s = 2 × apothem × tan(180° ÷ n)

Once you have s, compute the perimeter as n × s, then apply the formula.

How does this relate to circle area?

As the number of sides n increases, a regular polygon approaches a circle. The apothem approaches the radius r, and the perimeter approaches the circumference 2πr. Substituting into Area = ½ × perimeter × apothem gives:

Area ≈ ½ × 2πr × r = πr²

This is why the area formula for a circle, πr², has the same structure as the formula for a regular polygon — a regular polygon is just a circle with a very large (but finite) number of sides.

Frequently asked questions

Q: Does the area of a regular polygon formula appear on the GCSE formula sheet?

No — the formula A = ½ × perimeter × apothem and the trigonometric method for finding the apothem are not on the formula sheet. However, the area of a triangle (½ × base × height) and the trigonometric ratios are, so you can build up to the polygon area from those given formulae.

Q: What is the quickest method for a regular hexagon in particular?

A regular hexagon can be split into 6 equilateral triangles. If the side length is s, the area of one equilateral triangle is (√3/4)s², so the total hexagon area is 6 × (√3/4)s² = (3√3/2)s². This avoids needing tan and is worth memorising for the hexagon specifically.

Q: What if the polygon is not regular?

For irregular polygons you cannot use these formulae. Divide the shape into triangles and find the area of each separately, or use the coordinate geometry "shoelace formula" if coordinates are given. The splitting-into-triangles approach is the most reliable for any polygon.

Q: Why is it ½ × perimeter × apothem and not just perimeter × apothem?

Because you are summing the areas of triangles, each of which is ½ × base × height. The base of each triangle is one side of the polygon (length s) and the height is the apothem. Summing n triangles: n × ½ × s × apothem = ½ × (n × s) × apothem = ½ × perimeter × apothem. The ½ comes from the triangle formula.

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