Area of Regular Polygons

A regular polygon is a shape where every side has the same length and every interior angle is equal. Equilateral triangles, squares, regular pentagons, hexagons, and octagons are all regular polygons. These shapes appear everywhere — from honeycomb cells to stop signs to decorative floor tiles — and calculating their area is a practical skill in construction, design, and engineering.

While you can find the area of triangles and rectangles with simple base-times-height formulas, regular polygons with five or more sides need a different approach. The key concept is the apothem, and once you understand it, a single formula covers every regular polygon.

The Apothem

The apothem of a regular polygon is the perpendicular distance from the center of the polygon to the midpoint of any side. Think of it as the “inradius” — the radius of the largest circle that fits inside the polygon, touching every side.

For a regular polygon with $n$ sides of length $s$, the apothem is:

$a = \frac{s}{2\tan\!\left(\dfrac{\pi}{n}\right)}$

The apothem is always shorter than the circumradius (the distance from the center to a vertex). As the number of sides increases, the apothem gets closer to the circumradius, and the polygon looks more like a circle.

The General Area Formula

Every regular polygon can be divided into $n$ identical isosceles triangles, each with a base equal to the side length $s$ and a height equal to the apothem $a$. Adding up those triangles gives the general formula:

$A = \frac{1}{2} \times \text{perimeter} \times \text{apothem} = \frac{1}{2} \times n \times s \times a$

This formula works for any regular polygon — a triangle ($n = 3$), a square ($n = 4$), a pentagon ($n = 5$), a hexagon ($n = 6$), and beyond. The perimeter is simply $n \times s$.

The diagram above shows how the apothem connects the center to the midpoint of a side at a right angle. The dashed lines from the center to two adjacent vertices outline one of the six identical triangles that make up the hexagon.

Area of a Regular Hexagon

The regular hexagon is the most commonly searched polygon area formula. A hexagon has 6 sides, and its apothem is $a = \frac{s\sqrt{3}}{2}$. Substituting into the general formula:

$A = \frac{1}{2} \times 6s \times \frac{s\sqrt{3}}{2} = \frac{3\sqrt{3}}{2}\,s^2$

$\boxed{A_{\text{hexagon}} = \frac{3\sqrt{3}}{2}\,s^2 \approx 2.598\,s^2}$

Hexagons are nature’s favorite shape — honeycombs, basalt columns, and turtle shells all use hexagonal geometry because hexagons tile a plane with zero wasted space while enclosing the maximum area for a given perimeter among all tileable shapes.

Example 1: Find the area of a regular hexagon with side length 8 cm.

$A = \frac{3\sqrt{3}}{2}(8)^2 = \frac{3\sqrt{3}}{2}(64) = \frac{192\sqrt{3}}{2} = 96\sqrt{3}$

$A = 96 \times 1.7321 \approx 166.3 \text{ cm}^2$

Answer: The area is approximately 166.3 square centimeters.

Example 2: A hexagonal patio tile has an apothem of 6 inches. Find its area.

If the apothem is 6 in, we first find the side length. For a hexagon, $a = \frac{s\sqrt{3}}{2}$, so:

$s = \frac{2a}{\sqrt{3}} = \frac{2(6)}{\sqrt{3}} = \frac{12}{\sqrt{3}} = \frac{12\sqrt{3}}{3} = 4\sqrt{3} \approx 6.928 \text{ in}$

The perimeter is $P = 6s = 6(4\sqrt{3}) = 24\sqrt{3} \approx 41.57$ in. Now apply the general formula:

$A = \frac{1}{2} \times P \times a = \frac{1}{2}(24\sqrt{3})(6) = 72\sqrt{3} \approx 124.7 \text{ in}^2$

Answer: The area is approximately 124.7 square inches.

Area of a Regular Pentagon

A regular pentagon (5 sides) has the area formula:

$\boxed{A_{\text{pentagon}} = \frac{1}{4}\sqrt{5(5 + 2\sqrt{5})}\;s^2 \approx 1.720\,s^2}$

This comes from the general formula with $n = 5$. The apothem of a regular pentagon is $a = \frac{s}{2\tan(36°)} \approx 0.6882\,s$.

Example 3: Find the area of a regular pentagon with side length 10 m.

$A = \frac{1}{4}\sqrt{5(5 + 2\sqrt{5})} \times (10)^2$

First, compute the constant:

$2\sqrt{5} \approx 4.4721$

$5 + 4.4721 = 9.4721$

$5 \times 9.4721 = 47.3605$

$\sqrt{47.3605} \approx 6.8819$

$\frac{6.8819}{4} \approx 1.7205$

$A \approx 1.7205 \times 100 = 172.05 \text{ m}^2$

Answer: The area is approximately 172.1 square meters.

Area of a Regular Octagon

A regular octagon (8 sides) has the area formula:

$\boxed{A_{\text{octagon}} = 2(1 + \sqrt{2})\,s^2 \approx 4.828\,s^2}$

The apothem of a regular octagon is $a = \frac{s}{2\tan(22.5°)} \approx 1.2071\,s$.

Octagons are familiar as stop signs — every standard stop sign in the United States is a regular octagon.

Example 4: A stop sign has sides that are 12.4 inches long. Find its area.

$A = 2(1 + \sqrt{2})(12.4)^2$

$A = 2(1 + 1.4142)(153.76)$

$A = 2(2.4142)(153.76)$

$A = 4.8284 \times 153.76$

$A \approx 742.4 \text{ in}^2$

Answer: The area of the stop sign is approximately 742.4 square inches (about 5.16 square feet).

Example 5: An octagonal gazebo has an apothem of 6 feet. Find its area.

First, find the side length from the apothem. For an octagon, $a = \frac{s}{2\tan(22.5°)}$, so:

$s = 2a\tan(22.5°) = 2(6)(0.41421) = 4.971 \text{ ft}$

The perimeter is $P = 8s = 8(4.971) = 39.77$ ft. Now apply the general formula:

$A = \frac{1}{2} \times P \times a = \frac{1}{2}(39.77)(6) = 119.3 \text{ ft}^2$

Answer: The area of the gazebo floor is approximately 119.3 square feet.

Formula Reference Table

As the number of sides increases, the multiplier grows and the polygon approaches a circle with area $\pi r^2$.

Real-World Applications

Hexagonal Tiles

Hexagonal tiles are popular in kitchens and bathrooms because they tessellate perfectly — they cover a surface with no gaps and no overlaps. To estimate how many tiles you need, calculate the area of one tile and divide it into the total floor area.

Example: A bathroom floor is 60 square feet. Each hexagonal tile has a side length of 4 inches. How many tiles are needed?

Area of one tile: $A = \frac{3\sqrt{3}}{2}(4)^2 = \frac{3\sqrt{3}}{2}(16) = 24\sqrt{3} \approx 41.57 \text{ in}^2$

Convert the floor area: $60 \text{ ft}^2 \times 144 \text{ in}^2/\text{ft}^2 = 8640 \text{ in}^2$

Number of tiles: $\frac{8640}{41.57} \approx 208$ tiles

In practice, order 10-15% extra for cuts along walls and waste.

Octagonal Stop Signs

The standard US stop sign is a regular octagon with a flat-to-flat width of 30 inches. Knowing the area helps when calculating material costs for sign manufacturing or reflective sheeting.

Pentagonal Structures

Regular pentagons appear in gazebo designs, decorative windows, and architectural details. The Pentagon building in Arlington, Virginia, is the most famous pentagonal structure, though its shape is a concentric set of pentagons rather than a single regular pentagon.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• A regular polygon has all sides equal and all angles equal
• The apothem is the perpendicular distance from the center to the midpoint of a side
• The universal formula $A = \frac{1}{2} \times \text{perimeter} \times \text{apothem}$ works for every regular polygon
• Hexagon: $A = \frac{3\sqrt{3}}{2}\,s^2 \approx 2.598\,s^2$
• Pentagon: $A = \frac{1}{4}\sqrt{5(5+2\sqrt{5})}\,s^2 \approx 1.720\,s^2$
• Octagon: $A = 2(1+\sqrt{2})\,s^2 \approx 4.828\,s^2$
• As the number of sides increases, a regular polygon approaches a circle — and its area approaches $\pi r^2$
• These formulas apply to real-world objects: hexagonal tiles, octagonal stop signs, and pentagonal architectural features

Return to Geometry for more topics in this section.