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Area of a Pentagon

A polygon with five sides is called a pentagon. It can be a simple polygon or a self-intersecting one. In a regular pentagon, the five sides are equal in length and the internal angles are equal – 1080. The exterior angles are 720. In an irregular pentagon, this is not the case- the sides may not be equal and the angles can be different. There can be five diagonals in a regular pentagon. A convex pentagon is one in which the five vertices point outward. In a concave pentagon, the opposite happens. Let us learn the various methods to find out the area of a pentagon.

Image Credits: Tutorvista

The apothem of a pentagon

The apothem is a line from the centre of the pentagon drawn to the midpoint of a side such that it is perpendicular to the side. (this is different from the radius. The radius extends from the centre to the vertex.)

Calculating the area of a regular pentagon.

  • The Pentagon is divided into five triangles, by drawing five radiuses from the centre to radiate out to each of the vertices.
  • Now, the apothem is drawn, which is the height of each triangle.
  • Thus, the area of each triangle is calculated by A = ½ * base * height, where base = side length of the Pentagon and height = length of the apothem of the triangle.
  • Area of Pentagon = perimeter × apothem / 2
  • The small triangle is right-angled and so we can use sine, cosine and tangent to find how the side, radius, apothem and 5 (number of sides) are related:
sin(π/n) = (Side/2) / Radius Side = 2 × Radius × sin(π/5)
cos(π/n) = Apothem / Radius Apothem = Radius × cos(π/5)
tan(π/n) = (Side/2) / Apothem  → Side = 2 × Apothem × tan(π/5)

If only the side length is known and apothem is not known, we proceed differently.

  • Divide the pentagon into triangles.
  • Each triangle is subdivided into two right-angled triangles.
  • The angle of the centre of the pentagon is 360 (3600/10).
  • Thus, tan 36 = base/height.
  • Here, base = ½ of the length of the side of the pentagon.
  • Thus, height, or length of apothem = base length/ tan 36 = (side length/2)/ tan 36
  • Now if height is known, area of the triangle = ½ * base * height = ½ * side * apothem.
  • Thus, area of octagon = 5*1/2*s*a = (5*s2)/(4*tan 360) = (5s2) / (4√(5-2√5)).

If only the perimeter is known

  • P = 5s
  • S= p/5 where p = perimeter and s = length of the side.
  • Area of the pentagon = ½* p * a where a = apothem length.

If only the radius is known,

  • Area = (5/2)r2sin(72º), where r is the radius.

If it is an irregular pentagon, the easiest way is to divide it into a number of geometric figures, right angled triangles, squares or otherwise, and then proceed using appropriate formulas.

Examples

Question 1: Find the area of a pentagon of side 5 cm and apothem length 3 cm.
Solution:

Given,
s = 5 cm
a = 3 cm

Area of a pentagon
= 5/2 s*a

= 5/2 * 5 * 3 cm2

= 75/2 cm2

= 37.5 cm2
Question 2: Find the area of a pentagon of side 12 cm and apothem height 7 cm.
Solution:

Given,
s = 12 cm
r = 7 cm

Area of a pentagon
= 5 / 2 * s *a

= 5/2 * 12 * 7 cm2

= 5∗12∗7 /2 cm2

= 420 / 2 cm2

= 210 cm2

 

Question: Find the area of a pentagon of side 10 cm and apothem length 5 cm.
Solution:
Given,
s = 10 cm
a = 5 cm

Area of a pentagon
= 5 / 2 *  s *a
= 5 / 2 × 10 × 5 cm2
= 125 cm2

 Consider one triangle formed by joining two of the adjacent vertices with the center of a pentagon.

The interior angle O = 360o5 = 72º.

Since the triangle AOB is an isosceles triangle (AO = BO) ≤ A =≤ B = n.

So the measure of angle A = >
In triangle ∆ AOB = 72º + n + n = 180º
2n = 180º – 72º
2n = 108º.

n = 108o2 = 54º.

Now use the trigonometric ratio to get the value of h (height of the triangle which is same as the apothem of the pentagon.)

tan A = h(half of AB)

So h = 12 s × tan 54º

= 0.3369 s

So area of the triangle = 12 s h = 12 s ×(0.3369 s)

=0.16845 s2
So we can represent the area of a regular pentagon only in terms of its side length

= 5 × Area of each triangle
= 5 × (0.16845 s2)
= 0.84225 s2

Question: Find the area of the given regular pentagon whose each side measures 8 cm.
Solution:
Apply the formula for area with side using trigonometry

A = 0.84225 s2
= 0.84225 × (8)2

= 53.094 sq cm.

Check out our article on Geometry formulas here.

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