Question No. 21
What is the ratio of the area of a regular 12 sided polygon to the area of a regular octagon, if both the polygons are inscribed in the same circle?
Answers
Answer:
Ratio = 3/2.82
Step-by-step explanation:
A general formula for the area of a polygon having n number of sides which is inscribed inside a circle of radius r is given as;
Area = ½ x n x r2 sin(2π/n)
The value of n is 12 for a 12 sided polygon, putting this value;
Area of polygon = ½ x 12 x r2 sin(2π/12)
Area of polygon = 6 x r2 sin(30)
Area of polygon = 6 x r2 x 0.5
Area of polygon = 3 r2
The value of n is 8 for octagon, putting this value;
Area of octagon = ½ x 8 x r2 sin(2π/8)
Area of octagon = 4 x r2 sin(45)
Area of octagon = 4 x r2 x 0.707
Area of octagon = 2.82 r2
Ratio of Area of polygon to octagon = 3 r2/2.82r2
Ratio = 3/2.82
The ratio of the area of a regular polygon to the area of a regular octagon is 1.5 : 1.4
Step-by-step explanation:
The area of the polygon inscribed within circle is given by the formula:
A₁ = 1/2 × n × r² × sin(2π/n)
Where,
n = Number of sides = 12
On substituting the values, we get,
∴ A₁ = 1/2 × 12 × r² × sin(2π/12) = 6 × r² × sin(π/6)
The area of the octagon inscribed within circle is given by the formula:
A₂ = 1/2 × n × r² × sin(2π/n)
Where,
n = Number of sides = 8
On substituting the values, we get,
∴ A₂ = 1/2 × 8 × r² × sin(2π/8) = 4 × r² × sin(π/4)
From question, the circle is same. Thus,
6 × r² × sin(π/6) : 4 × r² × sin(π/4)
3 × sin(π/6) : 2 × sin(π/4)
3 × 0.5 : 2 × 0.70
∴ 1.5 : 1.4