Pentagon Calculator

Created by Luis Hoyos
Last updated: Jul 09, 2022

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.

Formulas for pentagon area, apothem, perimeter, and many more

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

Area

a²/4 × √(25 + 10√5)

Perimeter

5a

Diagonal (d)

a/2 × (1 + √5)

Height (h)

a/2 × √(5 + 2√5)

Circumcircle radius (R)

a/10 × √(50 + 10√5)

Incircle radius (apothem) (r)

a/10 × √(25 + 10√5)

Image of a pentagon with side, diagonal, height, circumcircle radius, and apothem marked.
Image of a pentagon with side, diagonal, height, circumcircle radius, and apothem marked.

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.

How to find the area of a pentagon with 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 aa:

    a=10r25+105a = \frac{10r}{\sqrt{25 + 10\sqrt5}}

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

    area=a2425+105\text{area} = \frac{a^2}{4} \sqrt{25 + 10\sqrt5}

    area=14(10r25+105)225+105\text{area} = \frac{1}{4}\left(\frac{10r}{\sqrt{25 + 10\sqrt5}}\right)^2 \sqrt{25 + 10\sqrt5}

    area=14(100r225+105)\text{area} = \frac{1}{4}\left(\frac{100r^2}{\sqrt{25 + 10\sqrt5}}\right)

    area=25r225+105\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)

Luis Hoyos
image of pentagon with side, diagonal, height, circumcircle radius and apothem marked.
Side (a)
in
Perimeter
in
Area
in²
Diagonal (d)
in
Height (h)
in
Circumcircle radius (R)
in
Incircle radius (apothem) (r)
in
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