Area of Regular Polygons: Apothem Formula

The area of a regular polygon is half the apothem times the perimeter: $A = \tfrac{1}{2}aP$ . The apothem $a$ is the distance from the center to the middle of a side. A regular polygon has all sides equal and all angles equal, like a stop sign or a honeycomb cell.

Before this lesson: You should know Interior and Exterior Angles of Polygons, Special Right Triangles and Trigonometric Ratios.

Parts of a Regular Polygon

A regular polygon has all sides congruent and all angles congruent. Every regular polygon has a center, the point that is the same distance from every vertex.

The radius goes from the center to a vertex.
The apothem goes from the center to the midpoint of a side. It is perpendicular to that side.
A central angle is formed by two radii drawn to neighboring vertices. For a regular polygon with $n$ sides, it measures $\frac{360^\circ}{n}$ .

Regular hexagon with center O and side length 6. A radius goes from O to a vertex. The apothem goes from O to the midpoint of a side and meets that side at a right angle; it is 3 times the square root of 3. The radius, apothem and half a side form a 30-60-90 right triangle. The central angle between two neighboring radii is 60 degrees. — In a regular hexagon with side 6, the apothem is 3√3, so the area is ½(3√3)(36) = 54√3.

The Area Formula

Draw a radius to every vertex. This cuts a regular $n$ -gon into $n$ congruent triangles. Each triangle has base $s$ (one side) and height $a$ (the apothem). So each one has area $\tfrac{1}{2}as$ (see area of triangles).

A = n \cdot \tfrac{1}{2}as = \tfrac{1}{2}a(ns) = \tfrac{1}{2}aP

Here $P = ns$ is the perimeter. The formula works for every regular polygon. For a square with side $s$ , the apothem is $\frac{s}{2}$ and $P = 4s$ , so $A = \tfrac{1}{2} \cdot \frac{s}{2} \cdot 4s = s^2$ , as expected.

How to Find the Area of a Regular Polygon

Find the perimeter: number of sides times side length.
Find the apothem. If it is not given, draw the right triangle made by the apothem, half a side and a radius.
In that triangle, the angle at the center is half the central angle, $\frac{180^\circ}{n}$ . Use special right triangles or trig to find the apothem.
Use $A = \tfrac{1}{2}aP$ . Keep exact values or full calculator values until the end.

Calculator note: For polygons other than triangles, squares and hexagons, use the tangent ratio in degree mode. Type the whole expression, like 0.5 × (2.5 ÷ tan(22.5)) × 40, so you round only once.

Worked Examples

Example 1: Apothem given

A regular hexagon has side length 6 and apothem $3\sqrt{3}$ . Find its area.

Find the perimeter: $P = 6 \cdot 6 = 36$
Write the formula: $A = \tfrac{1}{2}aP$
Substitute: $A = \tfrac{1}{2}(3\sqrt{3})(36)$
Multiply: $A = 54\sqrt{3}$
Use a calculator: $A \approx 93.53$

Answer: $54\sqrt{3}$ square units, or about 93.53

Example 2: A pentagon

A regular pentagon has side length 10 cm and apothem 6.88 cm. Find its area.

Find the perimeter: $P = 5 \cdot 10 = 50$
Substitute into $A = \tfrac{1}{2}aP$ : $A = \tfrac{1}{2}(6.88)(50)$
Multiply: $A = 172$

Answer: about 172 cm² (the apothem was already rounded)

Example 3: Find the apothem with special right triangles

A regular hexagon has side length 8. Find its area.

A hexagon’s central angle is $360^\circ \div 6 = 60^\circ$ . The apothem splits it into two 30° angles, so the right triangle is a 30-60-90 triangle.
The short leg is half a side: $8 \div 2 = 4$
The apothem is the long leg, $\sqrt{3}$ times the short leg (see special right triangles): $a = 4\sqrt{3}$
Find the perimeter: $P = 6 \cdot 8 = 48$
Substitute: $A = \tfrac{1}{2}(4\sqrt{3})(48)$
Multiply: $A = 96\sqrt{3}$
Use a calculator: $A \approx 166.28$

Answer: $96\sqrt{3} \approx 166.28$ square units

Example 4: Find the apothem with trig

A regular octagon has side length 5. Find its area to the nearest hundredth.

Regular octagon with side 5 and center O. The central angle for one side is 45 degrees. The apothem splits it into two 22.5 degree angles and cuts the side into halves of 2.5. In the right triangle, tangent of 22.5 degrees equals 2.5 over the apothem. — The apothem, half a side and a radius form a right triangle with a 22.5° angle at the center.

Find the central angle: $360^\circ \div 8 = 45^\circ$
Halve it for the right triangle: $45^\circ \div 2 = 22.5^\circ$
The half side, 2.5, is opposite 22.5° and the apothem is adjacent, so use tangent: $\tan 22.5^\circ = \frac{2.5}{a}$
Solve for the apothem: $a = \frac{2.5}{\tan 22.5^\circ} \approx 6.04$
Find the perimeter: $P = 8 \cdot 5 = 40$
Substitute the exact apothem: $A = \tfrac{1}{2} \cdot \frac{2.5}{\tan 22.5^\circ} \cdot 40$
Use a calculator: $A \approx 120.71$

Answer: about 120.71 square units. Using the rounded apothem 6.04 would give 120.80, which is slightly off.

Example 5: Word problem with a gazebo

A gazebo floor is a regular hexagon with sides 4 ft long. How much floor area is there?

The short leg of the 30-60-90 triangle is half a side: $4 \div 2 = 2$
The apothem is the long leg: $a = 2\sqrt{3}$
Find the perimeter: $P = 6 \cdot 4 = 24$
Substitute: $A = \tfrac{1}{2}(2\sqrt{3})(24)$
Multiply: $A = 24\sqrt{3}$
Use a calculator: $A \approx 41.57$

Answer: about 41.57 ft²

Common Mistakes

Mistake	Example of the error	Fix
Using the radius as the apothem	Using 6 (the radius of a hexagon with side 6) for $a$	The apothem goes to the middle of a side: $a = 3\sqrt{3}$ .
Forgetting the ½	$A = aP = 108\sqrt{3}$	$A = \tfrac{1}{2}aP = 54\sqrt{3}$ .
Using the full central angle in the right triangle	$\tan 45^\circ = \frac{2.5}{a}$ for an octagon	The apothem halves the central angle: use 22.5°.
Using a whole side in the right triangle	$\tan 22.5^\circ = \frac{5}{a}$	The apothem halves the side: use 2.5.
Rounding the apothem too early	$\tfrac{1}{2}(6.04)(40) = 120.80$	Keep the full value: 120.71.

Related Idea: Regular Polygons and Circles

As the number of sides grows, a regular polygon looks more and more like a circle. The apothem gets close to the radius, and the perimeter gets close to the circumference. Then $\tfrac{1}{2}aP$ becomes $\tfrac{1}{2} \cdot r \cdot 2\pi r = \pi r^2$ , the area of a circle.

Practice Problems

Round decimal answers to the nearest hundredth.

A regular hexagon has side 10 and apothem $5\sqrt{3}$ . Find its area.
A regular pentagon has perimeter 40 in and apothem 5.5 in. Find its area.
A regular decagon has side 4 cm and apothem 6.16 cm. Find its area.
An equilateral triangle has side 12. Its apothem is $2\sqrt{3}$ . Find its area.
A regular hexagon has apothem 6. Find its side length and area.
Find the central angle of a regular nonagon (9 sides).
A regular octagon has side 10. Find its area.
A regular pentagon has side 6. Find its area.
A square has apothem 7. Find its area two ways.
A stop sign is a regular octagon that is 30 in across from one flat side to the opposite side. Find its area.

Show answers

Find the perimeter: $P = 6 \cdot 10 = 60$
Substitute: $A = \tfrac{1}{2}(5\sqrt{3})(60)$
Multiply: $A = 150\sqrt{3}$
Use a calculator: $A \approx 259.81$
Substitute: $A = \tfrac{1}{2}(5.5)(40)$
Multiply: $A = 110$ in²
Find the perimeter: $P = 10 \cdot 4 = 40$
Substitute: $A = \tfrac{1}{2}(6.16)(40)$
Multiply: $A = 123.2$ cm²
Find the perimeter: $P = 3 \cdot 12 = 36$
Substitute: $A = \tfrac{1}{2}(2\sqrt{3})(36)$
Multiply: $A = 36\sqrt{3}$
Use a calculator: $A \approx 62.35$ . This matches $\frac{\sqrt{3}}{4}(12)^2 = 36\sqrt{3}$ .
The apothem is the long leg of a 30-60-90 triangle, so the short leg is $\frac{6}{\sqrt{3}} = 2\sqrt{3}$
A side is twice the short leg: $s = 4\sqrt{3}$
Find the perimeter: $P = 6 \cdot 4\sqrt{3} = 24\sqrt{3}$
Substitute: $A = \tfrac{1}{2}(6)(24\sqrt{3})$
Multiply: $A = 72\sqrt{3}$
Use a calculator: $A \approx 124.71$
Divide: $360^\circ \div 9 = 40^\circ$
Central angle: $360^\circ \div 8 = 45^\circ$ , half is 22.5°
Half a side is 5: $\tan 22.5^\circ = \frac{5}{a}$
Solve: $a = \frac{5}{\tan 22.5^\circ}$
Find the perimeter: $P = 8 \cdot 10 = 80$
Substitute: $A = \tfrac{1}{2} \cdot \frac{5}{\tan 22.5^\circ} \cdot 80$
Use a calculator: $A \approx 482.84$
Central angle: $360^\circ \div 5 = 72^\circ$ , half is 36°
Half a side is 3: $\tan 36^\circ = \frac{3}{a}$
Solve: $a = \frac{3}{\tan 36^\circ}$
Find the perimeter: $P = 5 \cdot 6 = 30$
Substitute: $A = \tfrac{1}{2} \cdot \frac{3}{\tan 36^\circ} \cdot 30$
Use a calculator: $A \approx 61.94$
The side is twice the apothem: $s = 14$
Way 1: $A = s^2 = 14^2 = 196$
Way 2: $P = 4 \cdot 14 = 56$ , so $A = \tfrac{1}{2}(7)(56) = 196$
The distance across from side to side is two apothems: $a = 30 \div 2 = 15$
Half the central angle is 22.5°, so half a side is $15 \tan 22.5^\circ$
A side is $s = 30 \tan 22.5^\circ$
Find the perimeter: $P = 8 \cdot 30 \tan 22.5^\circ = 240 \tan 22.5^\circ$
Substitute: $A = \tfrac{1}{2}(15)(240 \tan 22.5^\circ) = 1800 \tan 22.5^\circ$
Use a calculator: $A \approx 745.58$ in²

FAQ

What is the apothem of a regular polygon?

It is the segment from the center to the midpoint of a side. It meets the side at a right angle, so it is the height of each triangle in the polygon.

What is the formula for the area of a regular polygon?

$A = \tfrac{1}{2}aP$ , where $a$ is the apothem and $P$ is the perimeter.

How do you find the apothem if it is not given?

Draw the right triangle made by the apothem, half a side and a radius. The angle at the center is $\frac{180^\circ}{n}$ . Use special right triangles for 3, 4 or 6 sides, or the tangent ratio for others.

What is the difference between the apothem and the radius?

The radius goes from the center to a vertex. The apothem goes from the center to the middle of a side, so it is shorter.

Summary

A regular polygon has equal sides and equal angles. Its central angle is $\frac{360^\circ}{n}$ .
The apothem goes from the center to the midpoint of a side, at a right angle.
Area: $A = \tfrac{1}{2}aP$ , where $P$ is the perimeter.
Find a missing apothem with a right triangle whose center angle is $\frac{180^\circ}{n}$ .
Round only at the end.

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