Answer first
What this calculator tells you
Calculate the area, perimeter, apothem and interior angle of a regular polygon from its sides and side length. Lay out a hexagonal patio, an octagonal gazebo or a stop sign from one side length. Formula: Area = n × s² ÷ (4 × tan(π ÷ n)); apothem = s ÷ (2 × tan(π ÷ n)); interior angle = (n − 2) × 180° ÷ n. At the worked-example inputs, the area is 65. Holding every other input steady, moving number of sides (n) from 2 to 10 moves the result from 25 to 192.4.
Transparent method
The formula
Lay out a hexagonal patio, an octagonal gazebo or a stop sign from one side length.
Worked example
Example inputs
How to interpret the result
A regular polygon splits into identical triangles around its center, and each triangle's height is the apothem. Six sides of 5 units give a perimeter of 30, an apothem of about 4.33 and an area near 65. The interior angle follows from the side count alone: 120 degrees for a hexagon, 135 for an octagon. As the sides multiply, the shape approaches a circle.
At the worked-example inputs the area is 65. It rises with number of sides (n) and side length (s).
Formulas assume ideal geometric shapes. Real-world measurements carry their own error, and every input must use the same unit: mixing feet and inches silently produces a wrong answer.
Before you rely on it
What to check
Confirm every side and every angle match. A shape with equal sides but different angles is a rhombus-like figure, and these formulas will not fit it.
The common error
Where people go wrong with regular polygon calculator
Using the distance from the center to a corner in place of the side. That is the circumradius, which is larger than the side for most polygons and inflates the area.
Sensitivity evidence
How number of sides (n) changes the area
Holding every other input at the worked-example value, moving number of sides (n) from 2 to 10 moves the area from 25 to 192.4: a spread of 167.4, or 258% of the worked-example result.
| Number of sides (n) | Area | Perimeter | Apothem | Interior angle (degrees) |
|---|---|---|---|---|
| 2 | : | : | : | : |
| 4 | 25 | 20 | 2.5 | 90 |
| 6worked example | 65 | 30 | 4.3 | 120 |
| 8 | 120.7 | 40 | 6 | 135 |
| 10 | 192.4 | 50 | 7.7 | 144 |
Every input, tested
Which input moves the area most
Of the 2 inputs, number of sides (n) moves the area most (95.7 across the range tested) and side length (s) moves it least (26).
| Input | Tested from | To | Area at each end | Swing |
|---|---|---|---|---|
| Number of sides (n) | 4 | 8 | 25 to 120.7 | 95.7 (147%) |
| Side length (s) | 4.5 | 5.5 | 52.6 to 78.6 | 26 (40%) |
Two variables at once
Area by number of sides (n) and side length (s)
| Number of sides (n) \ Side length (s) | 4 | 5 | 6 |
|---|---|---|---|
| 2 | : | : | : |
| 4 | 16 | 25 | 36 |
| 6 | 41.6 | 65 | 93.5 |
| 8 | 77.3 | 120.7 | 173.8 |
| 10 | 123.1 | 192.4 | 277 |
The highlighted cell is the worked example: 65.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Number of sides (n) | 6 | 3 or more. A hexagon has 6, an octagon has 8. |
| Side length (s) | 5 | Enter the side length (s) used in this calculation. |
| Area | 65 | |
| Perimeter | 30 | |
| Apothem | 4.3 | |
| Interior angle (degrees) | 120 | |
Inputs, definitions and assumptions
Number of sides (n)
3 or more. A hexagon has 6, an octagon has 8. The prefilled worked-example value is 6.
Side length (s)
Enter the side length (s) used in this calculation. The prefilled worked-example value is 5.
How to use this calculator
- 1Verify the inputs. Gather number of sides (n) and side length (s) from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the area at 65. Store your own version of it as Scenario A.
- 3Test one change. Start with number of sides (n), the input with the biggest effect here: moving number of sides (n) from 4 to 8 takes the area from 25 to 120.7, a swing of 147% of the worked-example figure.
- 4Check the extremes. At half the example number of sides (n) (3) the area is 10.8; at double (12) it is 279.9.
People also ask
Frequently asked questions
How do you calculate regular polygon?
Area = n × s² ÷ (4 × tan(π ÷ n)); apothem = s ÷ (2 × tan(π ÷ n)); interior angle = (n − 2) × 180° ÷ n. At the worked-example inputs the area is 65.
What does the regular polygon result mean?
Lay out a hexagonal patio, an octagonal gazebo or a stop sign from one side length. At the worked-example inputs the area is 65. It rises with number of sides (n) and side length (s).
How much does number of sides (n) change the area?
Holding every other input at the worked-example value, moving number of sides (n) from 2 to 10 moves the area from 25 to 192.4, a spread of 167.4.
What are the limits of this regular polygon calculator?
Formulas assume ideal geometric shapes. Real-world measurements carry their own error, and every input must use the same unit: mixing feet and inches silently produces a wrong answer. The tables on this page test number of sides (n) only from 2 to 10; a value outside that range is not tabulated here.
Which input moves the area most in the regular polygon calculator?
Ranked by how far each moves the area across the range tested: number of sides (n) (95.7, 147%) and side length (s) (26, 40%).
If I double number of sides (n) in the regular polygon calculator, does the area double?
Doubling it from 6 to 12 takes the area from 65 to 279.9, which is 4.31 times the worked-example figure. So the result grows faster than the input does. Halving it to 3 gives 10.8.
How much does side length (s) matter in the regular polygon calculator?
The worked example uses 5. With the other inputs left at the worked example, moving side length (s) from 4.5 to 5.5 takes the area from 52.6 to 78.6, a swing of 40% of the worked-example figure.
Which inputs change the perimeter in the regular polygon calculator?
At the worked-example inputs it is 30. Number of sides (n) takes it from 20 to 40 and side length (s) takes it from 27 to 33.
Which inputs change the apothem in the regular polygon calculator?
At the worked-example inputs it is 4.3. Number of sides (n) takes it from 2.5 to 6 and side length (s) takes it from 3.9 to 4.8.
Which inputs change the interior angle (degrees) in the regular polygon calculator?
At the worked-example inputs it is 120. Number of sides (n) takes it from 90 to 135.
How do I convert square feet to square yards?
Divide by 9, since a square yard is three feet by three feet. Cubic feet to cubic yards is a division by 27. For any unit, square the linear factor for area and cube it for volume.
Sources and evidence
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