Pentagon Calculator

With this pentagon calculator, you'll find essential properties of a regular pentagon: side, diagonal, height, perimeter, and area, as well as the circumcircle and incircle radius. Type any value, and the remaining parameters will be calculated on the spot.

If you are not sure what a pentagon is or how many sides a pentagon has, keep scrolling, and you'll find clarifying pictures with a short explanation.

Pentagon is a 5-sided polygon. A pentagon can be simple or self-intersecting.

The sum of the internal angles in a simple pentagon is 540°, so every internal angle is equal to 108°. A regular simple pentagon has all five sides equal in length. (In this article, we use the term "regular pentagon" to describe a regular simple pentagon).

Area A of a regular pentagon can be calculated from the formula:

area = a² × √(25 + 10√5) / 4, where a is a side of a regular pentagon.

Also, you can find the area having the circumscribed circle radius:

area = 5R² × √[(5 + √5)/2] / 4, where R is the circumcircle radius.

Perimeter P of a regular pentagon is equal to the side length multiplied by the number of vertices. A pentagon is a five-sided polygon, so the perimeter is:

perimeter = 5 × a

To calculate the height and diagonal of a regular pentagon, all you need to have given is the side length a:

• diagonal = a × (1 + √5) / 2

• height = a × √(5 + 2√5) / 2

A pentagon has five diagonals equal in length, which form a pentagram.

As we now know the pentagon definition, we can have a look at this step-by-step example:

1. Find out what is given. For a regular pentagon, one parameter is enough to find the remaining six.

2. Type the value into the pentagon calculator. Let's take the most famous, almost regular pentagon as an example – the Pentagon building, the headquarters of the US Department of Defense. From the Wikipedia page, we find out that it's 1414 ft wide – it's the height of the pentagram.

3. The pentagon parameters appear! They are:

   • Side – 918.9 ft;
   • Diagonal – 1486.8 ft;
   • Perimeter – 4594 ft (0.87 mi);
   • Area – 33.35 ac;
   • Circumcircle radius – 781.6 ft; and
   • Incircle radius – 632.4 ft.

Did you notice how enormous it is? Have a look at the perimeter – it's almost a mile! In reality, each side of the building is ~921 feet long – it looks like it's practically a regular pentagon!

If you are interested in other regular shapes, have a look at our great tools:

• Triangle calculator
• Square calculator
• Hexagon calculator
• Octagon calculator

To compute the area of a regular pentagon from side length, you need to apply the formula:

area = a² × √(25 + 10√5) / 4.

Plugging in a = 2, we obtain area = 2² × √(25 + 10√5) / 4 = √(25 + 10√5) ≈ 6.882.

To compute the apothem of a pentagon with side length a, apply the formula:

apothem = 0.5 × a / tan(π/5)

By simplifying tan, we obtain the following:

apothem = 0.1 × a × √(25 + 10√5 ).

To compute the internal angle of a pentagon:

1. Divide 360° by the number of sides: 360°/5 = 72°.

2. Subtract 72° from 180° to get the internal angle of a pentagon: 180° - 72° = 108°.

3. It follows that the sum of internal angles in a pentagon equals 5 × 108° = 540°.