A pentagon area calculator finds the area of a regular pentagon, a five-sided figure with equal sides and equal angles. Regular pentagons appear in geometry problems, in tiling and design, and in nature, such as in the cross-section of okra or in some star patterns. Because all sides are equal, the area can be found from a single side length, which makes the calculation compact.
A regular pentagon can be divided into five identical isosceles triangles meeting at the centre. Each triangle has a vertex angle of 72 degrees at the centre, because 360 degrees divided by 5 is 72, and the area of the pentagon is five times the area of one triangle. That division is the geometric reason behind the area formula, and it also explains why the tangent of 36 degrees appears: half of the central triangle gives a right triangle with a 36-degree angle at the centre.
What this calculator does
For a regular pentagon with side length s, area = (5 x s^2) / (4 x tan(36 degrees)). The tangent must be evaluated at 36 degrees, not radians, unless the calculator is set accordingly. This calculator accepts the side length and evaluates the formula directly, rounding to two decimal places. The result is in square units matching the side length: sides in centimetres give area in square centimetres.
An alternative formula uses the apothem, the distance from the centre to the midpoint of a side: area = (perimeter x apothem) / 2. For a regular pentagon, perimeter is 5s, so the two formulas agree when the apothem is known. The side-length formula is more convenient when only s is given, which is the common case in textbook problems.
Formula
Area = (5 x s^2) / (4 x tan(36 degrees)), where s is the side length of a regular pentagon.
How to use it
- Confirm the pentagon is regular, with all sides and all angles equal. This formula does not apply to an irregular pentagon.
- Enter the side length s, in the unit you want the area to follow.
- Square s, multiply by 5, and divide by 4 x tan(36 degrees).
- Read the result as square units, not linear units.
- Check the size: area grows with the square of side length, so doubling s should multiply area by 4.
Try the calculator
Worked example
Find the area of a regular pentagon with side length 10 units. First, s^2 = 100. Then 5 x 100 = 500. Next, tan(36 degrees) = 0.7265, so 4 x 0.7265 = 2.906. Finally, 500 / 2.906 = 172.05 square units. The area is therefore about 172.05 square units. As a scaling check, a side of 20 units would give 4 times that area, about 688.2 square units, because area scales with s^2.
If the side were 5 units instead, s^2 = 25, 5 x 25 = 125, and 125 / 2.906 = 43.01 square units. That is one quarter of the area for side 10, which matches the scaling rule and confirms the pattern.
How to check your answer
Test scaling: doubling the side should quadruple the area. If it does not, the formula was applied to a linear quantity instead of a squared one.
Compare with the apothem formula if the apothem is known. Both formulas must give the same area for a regular pentagon.
Check the angle unit. Using radians instead of degrees for 36 degrees gives a very different tangent and a wrong area.
Common mistakes
- Using the formula for an irregular pentagon, where sides differ and the centre division is not uniform.
- Evaluating tan(36) in radians instead of degrees.
- Forgetting to square the side length before multiplying.
- Reporting the result in linear units instead of square units.
Frequently asked questions
Does this formula work for any pentagon?
No. It works only for a regular pentagon, where all sides and angles are equal. An irregular pentagon needs to be divided into triangles or other shapes with known measurements.
What is the apothem of a pentagon?
The apothem is the distance from the centre to the midpoint of a side. For a regular pentagon, area also equals (perimeter x apothem) / 2.
Why does tan(36 degrees) appear?
Dividing a regular pentagon into five centre triangles gives a 72-degree angle at the centre. Halving one triangle for a right triangle gives 36 degrees, and its tangent relates half a side to the apothem.
