Pentagon Calculator

Welcome to our pentagon calculator. With this tool, you'll be able to:

• Calculate a pentagon area with one of its sides;
• Calculate a pentagon area with apothem; and
• Calculate the perimeter, diagonal, height, circumcircle radius, apothem.
• Many more!

It's important to clarify that this calculator only solves regular pentagons - the most common type of pentagon, in which all sides and internal angles are equal. Those internal angles equal 108°

You can look at our regular polygon calculator to solve for many other regular polygons, and if you need to learn more about area units, look at our area converter.

To calculate any pentagon characteristic, we only need to know its side length (a)

We can use the three isosceles triangles of the pentagon to deduct the area formula. The process consists of adding the triangle areas of those three triangles. The hard part is the algebraic manipulation. Do you dare to do it?

Now, let's see how to find the area of a pentagon with the apothem.

The formula for the area of a pentagon with the apothem is area = 25r²/√(25 + 10√5), where r is the apothem.

We can derive the formula by manipulating the previous equations.

1. First, let's solve the apothem formula for $a$:

   $a = \frac{10r}{\sqrt{25 + 10\sqrt5}}$

2. Now, let's replace a in the area formula

   $\text{area} = \frac{a^2}{4} \sqrt{25 + 10\sqrt5}$

   $\text{area} = \frac{1}{4}\left(\frac{10r}{\sqrt{25 + 10\sqrt5}}\right)^2 \sqrt{25 + 10\sqrt5}$

   $\text{area} = \frac{1}{4}\left(\frac{100r^2}{\sqrt{25 + 10\sqrt5}}\right)$

   $\text{area} = \frac{25r^2}{\sqrt{25 + 10\sqrt5}}$

3. That's it! The formula for the area of a pentagon with apothem is area = 25r²/√(25 + 10√5)