DWG NO. 10.3 — Lesson 3 of 5

Area of Regular Polygons

Unit 10: Area of Polygons and Circles · ~15–30 min

Objective

Find the area of a regular polygon using its apothem and perimeter.

Slicing a polygon into triangles

A regular polygon — equal sides, equal angles — can always be divided into congruent isosceles triangles by drawing segments from the center to each vertex. Each triangle's base is one side of the polygon, and its height is the apothem: the perpendicular distance from the center to a side.

Apothem (a): the perpendicular distance from the polygon's center to the midpoint of a side.
Regular polygon area: A = ½aP, where a is the apothem and P is the perimeter.

This comes directly from the triangle-slicing picture: each of the n congruent triangles has area ½(side)(a), and adding up n of those side lengths just gives the perimeter P — so the total is ½aP.

Segments from the center to each vertex slice the hexagon into congruent triangles with height a, the apothem.

Worked Example 1 · Regular hexagon

ProblemA regular hexagon has side length 8 cm and apothem 6.9 cm. Find its area.

1.Perimeter P = 6(8) = 48 cm.

2.A = ½aP = ½(6.9)(48) = ½(331.2).

Area ≈ 165.6 cm²

Worked Example 2 · Solve for the apothem

ProblemA regular pentagon has perimeter 40 in and area 110 in². Find its apothem.

1.A = ½aP, so 110 = ½(a)(40) = 20a.

2.a = 110 ÷ 20 = 5.5.

Apothem = 5.5 in

Guided practice

1.A regular octagon has side length 5 m and apothem 6.0 m. Find its area.
Show answer
P = 8(5) = 40 m. A = ½(6.0)(40) = ½(240) = 120 m².
2.A regular polygon has perimeter 60 ft and apothem 8 ft. Find its area.
Show answer
A = ½(8)(60) = ½(480) = 240 ft².
3.A regular hexagon has area 187.2 cm² and apothem 7.8 cm. Find its perimeter and side length.
Show answer
187.2 = ½(7.8)(P) = 3.9P, so P = 48 cm. Side = 48 ÷ 6 = 8 cm.
4.Explain why the apothem, not a side, must be used as the "height" in the regular polygon area formula.
Show answer
The polygon is built from n congruent triangles whose bases are the sides; the triangle-area formula needs the height perpendicular to that base, which is exactly the apothem — a side length wouldn't measure that perpendicular distance.
5.A regular square (n = 4) has side length 10 in. Using the apothem-perimeter formula, show it gives the same area as side².
Show answer
The apothem of a square is half the side, so a = 5. P = 40. A = ½(5)(40) = 100 in², which matches side² = 10² = 100 in².

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