Math logo


Area of a pentagon

There is no exact formula for calculating area of a pentagon. There are different ways to calculate area of a pentagon and I have chosen the one that I think is suitable for most people. Below is the info graphic and the explanation of how to calculate the area of a pentagon in 4 easy steps.

Area of a pentagon

Divide the pentagon into triangles (figures 1, 2, and 3)

We know that a regular pentagon has 5 sides equal in length and 5 interiors angles equal in measure. Each angle is 108 degrees.

A regular pentagon can be inscribed in a circle which means that F is the center of the circle and the center of the pentagon. Based on this math rule we now know the following:
AF = BF = CF = DF = EF
The above equality is true because all these segments are the radius of the circle. Now we have 5 identical isosceles triangles.

To find the area of the pentagon all we have to do is to find the area of one triangle (for this example we will use the triangle AFE).

Since AFE is an isosceles triangle, FG is the height of the triangle. FG is also perpendicular on the base AE and it divides the base AE into 2 equal segments.
Knowing the length of one side (for this example I have chosen to be 6) we can determine the length of AG and GE which is a half of 6 (AG = GE = AE/2 = 6/2 = 3). Now we can work better using a right angled triangle.

Calculating the angles measures (figure 4)

We know that each interior angle of a regular pentagon is 108 degree since the sum of all angles of a regular pentagon is 540 degree and all of the 5 angles are equal in measure. Dividing 540 by 5 we obtain 108. Angle a is a half of angle a1 which means a = 54 degree.

Finding angle f is an easy task knowing the other 2 angles of the triangle since the sum of all angles of a triangle must be 180 degree.
angle a + angle g + angle f = 180 degree
54 degree + 90 degree + angle f = 180 degree
angle f = 180 degree - 54 degree - 90 degree
angle f = 36 degree

Calculating the length of the sides (figure 5)

Now we have to find the length of the side FG. We know that AG = 3 and we know the measure of all angles. According to math rules we have:
FG = tan(a) x AG
You can easily calculate tan(a) using an online calculator.
tan(54) = 1.37638192 so FG = 1.37638192 × 3 = 4.12914576. For the sake of simplicity FG = 4.129

Calculating the area of the pentagon (figure 6)

Knowing the base and the height of a triangle we can now calculate first the area of the triangle then the area of the pentagon. The area of the triangle AFG is base multiply by height then divided by 2.
That means area of the triangle AFG is 3 multiply by 4.129 then divided by 2. The result is 6.193 square units. The pentagon is formed by 10 such identical triangles so to find the area of the pentagon we just need to multiply 6.193 by 10. The area of the pentagon ABCDE is 61.93 square units.