Regular Polygon Area Calculator
Area, perimeter, and angle of a regular polygon.
What it does
Regular polygons show up in tiles, nuts, gazebos, and garden beds. When sides and angles repeat, one side length plus the side count fixes the whole figure. That is why a hexagon patio can be priced from a single tape pull once you know it is truly regular.
The tool above turns sides and side length into area, perimeter, and corner angle. A hexagon with 10 cm sides gives 259.81 cm2 and 60 cm around, with 120 degree corners. The imperial mirror at 3.9 in gives 39.52 in2 and 23.4 in, the same shape in other units. Corners stay in degrees either way.
The area formula divides the figure into n triangles from the center. Each triangle has base s and height set by pi/n, which is where the tangent enters. Perimeter is the simple part, n times s, while angle follows from the triangle angle sum. As n grows, area for a fixed span climbs toward the circle.
This page assumes equal sides and equal angles. It does not handle irregular plots, rounded corners, or material waste. Measure diagonals to confirm regularity on site before cutting expensive stock.
How to use
- Pick metric (cm) or imperial (in).
- Enter the count of sides from 3 to 12.
- Enter one side length.
- Read area first, then perimeter.
- Read the corner angle for cutting or layout.
- Double check n before ordering material.
Formula
- n
- Count of equal sides, 3 to 12.
- s
- Length of one side, in cm or in.
- angle
- Corner angle in degrees, same in both unit modes.
Regular means equal sides and equal angles, with n from 3 to 12 here. At n = 3 the formula gives the equilateral triangle; as n grows the figure nears a circle for the same span.
Worked example
Example: hexagon with 10 cm sides
Defaults: metric hexagon with 10 cm sides. Imperial mirror is 3.9 in sides. Angles need no units.
- Inputs (metric default): 6 sides, side 10 cm.
- Angle: (6 minus 2) times 180 / 6 is 120 degrees.
- Perimeter: 6 times 10 is 60 cm.
- Area: 6 times 100 / (4 times tan 30 deg) is 259.81 cm2.
- Imperial mirror (6 sides, 3.9 in) gives 39.52 in2, 23.4 in, 120 degrees. Tool shows the same.
Reference table
| Figure | Area | Perimeter |
|---|---|---|
| Hexagon 10 cm | 259.81 cm2 | 60 cm |
| Pentagon 8 cm | 110.11 cm2 | 40 cm |
| Octagon 5 cm | 120.71 cm2 | 40 cm |
| Square 10 cm | 100 cm2 | 40 cm |
| Triangle 10 cm | 43.3 cm2 | 30 cm |
Limits
- Regular figures only. Irregular needs other methods.
- Straight sides. Arcs and chamfers excluded.
- No waste or joints. Order extra for cuts.
- Sides 3 to 12. Larger n nears a circle.
Common mistakes
- Using diameter or radius as side length.
- Feeding degrees to tan without converting to radians.
- Mixing cm with inches in one calculation.
- Assuming a near regular patio is exactly regular.
- Forgetting that area squares when side doubles.
Try it yourself
Quick checks. Answers sit tucked under each question.
Pentagon, side 8 cm?
Pentagon angle is 540 / 5 is 108 degrees. Area is 5 times 64 / (4 times tan 36 deg) is 110.11 cm2. Perimeter is 40 cm.
Octagon, side 5 cm?
Octagon angle is 1080 / 8 is 135 degrees. Area is 8 times 25 / (4 times tan 22.5 deg) is 120.71 cm2. Perimeter is 40 cm.
Questions
What is the area formula?
Area = n times s squared / (4 times tan(pi/n)). Here 600 / 2.3094 is 259.81 cm2.
How is perimeter figured?
Perimeter = n times s. Here 6 times 10 is 60 cm.
How is one corner angle figured?
Angle = (n minus 2) times 180 / n. Here 720 / 6 is 120 degrees.
Does this work for irregular shapes?
No. Sides must be equal and corners equal. Irregular lots need triangulation or coordinates.
Do I use degrees or radians for tan?
Use radians in code: tan(pi/n). Degrees are for display only.
What happens if I double the side?
Area grows with the square of side. Double the side and area quadruples.