In Geometry, a polygon is a closed two-dimensional figure made up of straight lines. We know that a triangle is a polygon with the least number of sides. Basically, the polygon can be classified as a regular or irregular polygon and convex or concave polygon. A regular polygon is a polygon in which all the sides and interior angles are equal. In this article, we are going to learn the area of a regular polygon, its formula and solved examples in detail.
What is the Area of a Regular Polygon?
The area of a regular polygon is the space enclosed by the boundary of the regular polygon. In other words, the area of a regular polygon is the area that is enclosed by it. It is generally measured in square units, such as cm2, m2, ft2, and so on. Generally, the area of a polygon can be determined using different formulas, based on whether the polygon is regular or irregular.
Area of Regular Polygon Formula
The area of regular polygon formulas for some of the most commonly used polygons are as follows:
Area of Equilateral triangle = (√3a2) /4 square units
Where “a” is the side length of an equilateral triangle
Area of Square = a2 square units
Where “a” is the side length of square
Area of Regular Pentagon = (1/4) ×√[5(5+2√5)] ×a2 square units
Where “a” is the side length of the pentagon
Area of Regular Hexagon = [3√3a2]/2 Square units
Where “a” is the side length of the hexagon.
Area of Regular Polygon of N-sides Formula
If “n” is the number of sides of a polygon, then the formula to find the area of regular polygon of n sides is given by:
Area of Regular Polygon Formula = [l2n]/[4tan(180/n)] Square units
Where “l” is the side length of polygon
“n” is the number of sides of the polygon.
Also, check: Area of Regular Polygon Calculator
Area of Regular Polygon Inscribed in a Circle
The area of a regular polygon inscribed in a circle formula is given by:
Area of a regular polygon inscribed in a circle = (nr2/2) sin (2π/n) square units
Where “n” is the number of sides
“r” is the circumradius.
Area of Regular Polygon Problems and Answers
Go through the below problems to find the area of a regular polygon.
Example 1:
Find the area of a regular hexagon whose side length is 2 cm.
Solution:
Given: Side length, a = 2 cm
We know that the formula for the area of a regular hexagon is [3√3a2]/2 Square units
Substituting the value in the formula, we get
A = [3√3(2)2]/2
A = (12√3)/2 = 6√3
We know that √3 = 1.732
So, A = 6(1.732) = 10.392 cm2
Therefore, the area of a regular hexagon with a side length of 2 cm is 10.392 cm2.
Example 2:
Calculate the area of a regular polygon whose side length is 6 cm and the number of sides is 5.
Solution:
Given that, the number of sides,n = 5
Side length, l = 6 cm
We know that,
Area of Regular Polygon Formula = [l2n]/[4tan(180/n)] Square units
Now, substitute the values in the formula, we get
A = [(62(5)]/[4tan(180/5)] cm2
A = 180/[4tan (180/5)] cm2
A = 180/ [4(tan 36)] cm2
A = 180/[4(0.7265)] cm2
A = 180/2.906 cm2
A = 61.94 cm2
Hence, the area of a regular polygon is 61.94 cm2.
Example 3:
Find the area of regular pentagon inscribed in a circle whose circumradius is 4 cm.
Solution:
Given: Number of sides, n = 5
Circumradius, r = 4 cm.
We know that the area of a regular polygon inscribed in a circle = (nr2/2) sin (2π/n) square units.
Substituting the values, we get
A = (5(4)2/2) sin (2π/5) cm2
A = (5(16)/2) sin (360/5) cm2
A = 40 sin (72) cm2
A = 40(0.951) cm2
A = 38.04 cm2
Therefore, the area of a regular pentagon inscribed in a circle is 38.04 cm2.
Area of Regular Polygon Practice Questions
Solve the following problems:
- Find the area of a regular hexagon whose side length is 4 cm.
- Compute the area of a regular polygon of side length 6 cm, whose number of sides is 7.
- Determine the area of a regular pentagon whose side length is 8 cm.
- Find the area of a regular pentagon inscribed in a circle whose circumradius is 4 cm.
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Frequently Asked Questions on Area of a Regular Polygon
What is the area of a regular polygon?
The area of a regular polygon is the space enclosed by the boundary of the regular polygon. The area of a regular polygon is generally measured in square units.
What is the formula for the area of a regular polygon of n-sides?
The formula for the area of a regular polygon of n-sides is [l2n]/[4tan(180/n)] Square units, where l is the side length of the polygon and n is the number of sides.
What is the formula for the area of a regular polygon inscribed in a circle?
The area of a regular polygon inscribed in a circle formula is (nr2/2) sin (2π/n) square units, where n is the number of sides and r is the circumradius.
Does the area of a regular polygon directly depend on the exterior angle?
No, the area of a regular polygon does not directly depend on the exterior angle.
What is the difference between the perimeter and the area of a regular polygon?
The area of a regular polygon is the space occupied by it, whereas the perimeter of a regular polygon is the total length of the boundary of a regular polygon.
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