Question

ABCDEF is a regular hexagon with centre O (in the following figure). If the area of triangle OAB is 9 cm², find the area of :
(i)the hexagon and
(ii)the circle in which the hexagon is incribed.​

Answer 1

Answer:

The area of hexagon is  54 cm² and area of circle is 65.324 cm².

Step-by-step explanation:

Let ‘r’ be the radius of a circle and ‘a’ be the side of a equilateral triangle.

Given :

Area of ∆OAB (equilateral) =  9 cm²

Area of hexagon = 6 × Area of equilateral triangle  

[A Regular hexagon is made up of 6 equal triangles]

(i) Area of hexagon = 6 × 9 = 54 cm²

Area of hexagon = 54 cm²

(ii) Area of equilateral ∆ = √3/4 × a²

9 = √3/4 × a²

9 × 4 = √3a²

36 = √3a²

a² = 36/√3

Side² = 36/√3 cm

Radius of circle,r = Side of a hexagon  

[In regular hexagon inscribed in a circle,its side is equal to the radius of a Circle]

r² =  36/√3 cm

Area of circle,A = πr²

A = 22/7 × 36/√3

A = 22/7 × 36/1.732

A = (22 × 36) /(7 ×1.732)

[√3 = 1.732]

A = 792/12.124

A = 65.324 cm²

Area of circle = 65.324 cm²

Hence, the area of hexagon is  54 cm² and area of circle is 65.324 cm².

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Answer 2

Answer:

The area of hexagon is 54 cm² and area of circle is 65.324 cm².

Step-by-step explanation:

The area of hexagon is 54 cm² and area of circle is 65.324 cm².

Step-by-step explanation:

Let ‘r’ be the radius of a circle and ‘a’ be the side of a equilateral triangle.

Given :

Area of ∆OAB (equilateral) = 9 cm²

Area of hexagon = 6 × Area of equilateral triangle

[A Regular hexagon is made up of 6 equal triangles]

(i) Area of hexagon = 6 × 9 = 54 cm²

Area of hexagon = 54 cm²

(ii) Area of equilateral ∆ = √3/4 × a²

9 = √3/4 × a²

9 × 4 = √3a²

36 = √3a²

a² = 36/√3

Side² = 36/√3 cm

Radius of circle,r = Side of a hexagon

[In regular hexagon inscribed in a circle,its side is equal to the radius of a Circle]

r² = 36/√3 cm

Area of circle,A = πr²

A = 22/7 × 36/√3

A = 22/7 × 36/1.732

A = (22 × 36) /(7 ×1.732)

[√3 = 1.732]

A = 792/12.124

A = 65.324 cm²

Area of circle = 65.324 cm²

Hence, the area of hexagon is 54 cm² and area of circle is 65.324 cm².

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