Pentagon Calculator

What is a pentagon calculator?

The pentagon calculator works out the geometry of a regular pentagon — the five-sided shape with equal sides and equal angles — from a single measurement: the side length. Type in one side and the calculator returns the area, the perimeter, and the apothem at once. It is handy for students checking geometry homework, makers laying out a five-sided panel or tabletop, and anyone designing patterns that use the pentagon, a shape that appears everywhere from soccer balls to architecture.

Properties of a regular pentagon

A regular pentagon has five equal sides and five interior angles of 108 degrees each. The apothem is the distance from the center of the pentagon to the midpoint of any side; it is also the radius of the largest circle that fits inside the shape. Because every side is the same length, the perimeter is simply five times the side, and the area can be expressed directly in terms of the side using a single constant.

How does it work?

Enter the side length and the calculator fills in the rest. The perimeter is five times the side. The area uses the exact constant for a regular pentagon, and the apothem uses the constant that relates the center-to-edge distance to the side. All three outputs update immediately as you change the side.

Formulas

With side length $s$, the perimeter of a regular pentagon is five times the side:

The area is given by the exact formula:

The constant $\frac{1}{4}\sqrt{5\left(5 + 2\sqrt{5}\right)} \approx 1.72048$, so the area is roughly $1.72048\,s^2$.

The apothem (center to the midpoint of a side) is:

The factor $\frac{1}{2\tan(36^\circ)} \approx 0.68819$, so the apothem is about $0.68819\,s$.

Here $P$ is the perimeter, $A$ the area, $a$ the apothem, and $s$ the side length.

Examples

1. A regular pentagon with a side of 1 unit:

2. A regular pentagon with a side of 6 units:

Practical notes

• The apothem is useful when you need the area as $A = \tfrac{1}{2} \times P \times a$, a formula that works for any regular polygon.
• A pentagon does not tile the plane on its own, unlike the regular hexagon — see the hexagon calculator for a six-sided shape that does.
• For other regular shapes, the equilateral triangle calculator and the circle area calculator handle the three-sided and round cases.

FAQs

How do I find the area of a regular pentagon?

Square the side length and multiply by the constant $\frac{1}{4}\sqrt{5\left(5 + 2\sqrt{5}\right)} \approx 1.72048$. For a side of 6 the area is about $1.72048 \times 36 \approx 61.9372$.

What is the apothem of a pentagon?

The apothem is the distance from the center to the midpoint of a side. For a regular pentagon it equals $\frac{s}{2\tan(36^\circ)} \approx 0.68819\,s$, so a side of 1 gives an apothem of about $0.6882$.

What is the perimeter of a regular pentagon?

Because all five sides are equal, the perimeter is just five times the side: $P = 5s$. A side of 6 gives a perimeter of 30.