The area of a pentagon is a bit complex compared to lower sided shapes. Area of a regular pentagon = pa/2, where p = perimeter of the pentagon and a = the apothem (straight line from the centre of the pentagon to the side) of the pentagon. The area of irregular pentagon doesn’t have any formulae as such.
In geometry, a pentagon (from the Greek πέντε pente which means five and γωνία gonia which means angle) is a five-sided polygon.
Categories of Pentagons
In general pentagons are divided into two categories: Regular or Irregular; Convex or Concave.
A regular pentagon is one with all equal sides and equal angles, with each side making an interior angle 72° at the center of pentagon and the angle between two sides (exterior angle) measuring 108°.
An irregular pentagon is a shape that does not have equal sides and/or angles and no specified angles.
A convex pentagon is one whose vertices / where the sides meet, is pointing outwards. All the interior angles are less than 180.
Concave pentagon whose vertices / where the sides meet points inwards. All the angles are not less than 180.
Finding Area of Pentagon
Area of pentagon can be found in 2 ways.
Area of Pentagon from the Length of Side and Apothem
This method is for regular pentagons. Besides the side length, we need to find the “apothem” of the pentagon.
The apothem is a straight line from the centre of the pentagon to the side, intersecting the side at a 90º right angle.
Step 1: Draw five lines from the center of the pentagon to each vertex (corners).
Now we have five equilateral triangles.
Step 2: Calculate the area of the triangle using the formula \(\).
Step 3: Multiply the area of one triangle into 5 so that we can get the total area of the pentagon.
Area of Pentagon Using Formula
Area of a regular pentagon = pa/2 = 5sa/2
p = the perimeter
a = the apothem
s = side length.
Area of regular pentagon = (5s2) / (4tan(36º)) = (5s2) / (4√(5-2√5)).
Properties of Pentagon
- The sum of all the internal angles in a pentagon is 540°(108° *5)
- A pentagon can be simple or self-intersecting.
Question 1. Find the area of the regular pentagon whose side is 14.6 cm and apothem length is 10cm.
Solution. Given that,
Side(s) = 14.6 cm
Apothem(a) = 10 cm
Area(A) = (5 ⁄ 2) × s × a
= (5 ⁄ 2) × 14.6 × 10 cm2
= 365 cm2.
Question 2. Shyam was given a Pentagon of which area is 720 units square and with a side of 19 units. Help him in finding the length of the apothem of the Pentagon?
Solution. Given that,
Area of the Pentagon(A) = 720 units square.
Length of a side (s) = 19 units
We know that the area of the Pentagon(A) = 5/2 × s × a.
720 = 5/2 × 19 × a.
Apothem(a) = 15.15 units
Therefore, the length of the apothem of the Pentagon whose are is 720 unit square is 15.15 units.
Question 3. Find the area of a pentagon of side 10 cm.
Solution. Given side of pentagon(s) = 10 cm
Construct a triangle by joining two of the adjacent vertices with the center of a pentagon.
For pentagon height of the triangle and the apothem are same.
The interior angle O = 360º/5 = 72º.
Since the triangle AOB is an isosceles triangle (AO = BO).
In triangle ∆ AOB = 72º + x + x = 180º.
2x = 180º – 72º.
2x = 108º.
x = 108º/2 = 54º.
Now by using the trigonometric ratio to get the value of h.
tan A = h(½ AB)
So, h = (5) × tan 54º = 6.88.
Area of the triangle = ½ s h = ½10 ×(6.88) = 34.5 cm2.
Area of polygon = 5 × Area of each triangle = 5×34.5 = 172 cm2 .
Required area of the pentagon with side 10 cm is 172 cm2.
For finding the area of an irregular polygon we must first separate the irregular shape into regular polygons. We then use the regular polygon area formulas for finding the area of each of those polygons. At the end we will add all those areas together to get the total area of the irregular polygon.
A pentagon has five straight sides, but does not have to be of equal length but whereas a regular pentagon has five equal sides and five equal angles.
In a pentagon the length of each line is equal to the other ones and the angle between two lines is 108°, which gives a sum of inner angles 540°. In this case there are no parallel lines and it has 5 sides.