How to find the Area of a hexagon with radius?
In geometry, a hexagon is a two-dimensional closed shape with six sides. There are two kinds of hexagons based on the lengths of the sides and the angles they form, namely regular hexagons, irregular hexagons, convex hexagons, and concave hexagons. A regular hexagon is a hexagon with equal side lengths and each of its interior angles is 120Β°. An irregular hexagon is a hexagon whose measurements of side lengths and angles are different. A concave hexagon is a hexagon with at least one interior angle that is greater than 180Β°. A convex hexagon is a hexagon with no angle pointing inwards; i.e., the measure of every interior angle is less than 180Β°.
Area of a hexagon
The area of the hexagon is referred to as the area enclosed by its six sides.
The formula for the area of a regular hexagon is
The area of a regular hexagon =
- In a regular hexagon, the radius (r) of a hexagon is equal to the length of its side (a), i.e., a = r. Thus, the regular hexagon is divided into six equilateral triangles.
Now,
The area of a regular hexagon with radius βrβ =
Derivation
To derive the area of a regular hexagon, divide it into six triangles by joining its opposite vertices with a line segment.
Each triangle is an equilateral triangle with its side length βaβ and height β3a/2 (which is also the apothem of the hexagon).
We know that,
The area of a triangle = Β½ Γ base Γ height
=Β½ Γ a Γ (β3a/2)
The area of a triangle = β3a2/4
Since there are six equilateral triangles
The area of a regular hexagon = 6 Γ (Area of the equilateral triangle)
A = 6 Γ (β3a2/4)
A =
In a regular hexagon, the length of the side (a) is equal to the radius of the hexagon (r)
Hence,
The area of the regular hexagon =
Sample Problems
Problem 1: Find the area of a regular hexagon with a radius of 6 in.
Solution:
Given,
The radius of the regular hexagon (r) = 6 in
We know that the radius of a regular hexagon is equal to the length of each side of the hexagon.
(Length of the side of a regular hexagon) a = r
The area of the hexagon = 3β3r2/2
β A = 3β3(6)2/2
β A = 3β3(36)/2
β A = 54β3 sq. in
Hence, the area of the hexagon is 54β3 sq. in.
Problem 2: Determine the radius of the hexagon if its area is 96β3 sq. cm.
Solution:
Given,
The area of the hexagon = 96β3 sq. cm
We know,
The area of the hexagon = 3β3a2/2
β 96β3 = 3β3a2/2
β a2 = 96β3 Γ (2/3β3)
β a2 = 64 β a = β64
β a = 8 cm.
We know that the radius of a regular hexagon is equal to the length of each side of the hexagon, i.e., a = r = 8 cm
So, the radius of the regular hexagon (r) is 8 cm.
Problem 3: Find the area of the hexagon whose radius is 12 units.
Solution:
The radius of the regular hexagon (r) = 12 units
We know that the radius of a regular hexagon is equal to the length of each side of the hexagon.
(Length of the side of a regular hexagon) a = r
The area of the hexagon = 3β3a2/2
β A = 3β3(12)2/2
β A = 3β3(144)/2
β A = 216β3
β A = 374.112β¬β¬ square units
Hence, the area of the hexagon is 374.112β¬ square units.
Problem 4: Calculate the radius of the hexagon if its area is 24β3 sq. in.
Solution:
Given,
The area of the hexagon = 24β3 sq. cm
We know,
The area of the hexagon = 3β3a2/2
β 24β3 = 3β3a2/2
β a2 = 24β3 Γ (2/3β3)
β a2 = 16 β a = β16
β a = 4 cm.
We know that the radius of a regular hexagon is equal to the length of each side of the hexagon, i.e., a = r = 4 cm
So, the radius of the regular hexagon (r) is 4 cm.
Problem 5: Calculate the area of the hexagon if its perimeter is 54 cm.
Solution:
Given,
The perimeter of the hexagon = 54 cm.
We know that the perimeter of a hexagon = 6a
β 6a = 54 β a = 54/6
β a = 9 cm
We know that the radius of a regular hexagon is equal to the length of each side of the hexagon
Hence a = r = 6 cm.
The area of the hexagon = 3β3a2/2
A = 3β3(9)2/2
A = 3β3(81)/2 = 121.5β¬β3
A = 121.5 Γ 1.732 = 210.438β¬ sq. cm
Hence, the area of the hexagon is 210.438β¬ sq. cm.
Problem 6: Calculate the area of a regular hexagon whose radius is 3 units.
Solution:
The radius of the regular hexagon (r) = 3 units
The area of the hexagon = 3β3r2/2
β A = 3β3(3)2/2
β A = 3β3(9)/2
β A = (13.5)β3 square units
β A = 23.382β¬ square units
Hence, the area of the hexagon is 23.382β¬ square units.
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