Question

A regular hexagon is circumscribed about a circle with a radius of 6 cm. Find the area shaded.

Anyone help please ​

Answer 1

Solution :-

The regular hexagon is inscribed in a circle of radius 6cm.

So, it is inside the circle.

By joining opposite sides of hexagon, it forms 6 central angles at centre O each of which = 360/6 = 60° .

And six triangles are formed.

The two sides of each triangle are the radius of the circle and thus are equal.

∴ The base angles of every triangle are equal.

∵ Central angle is 60°.

→ Base angles = (120/2) = 60°

∴ The triangles are Equilateral Triangles.

→ All sides are equal.

∴ All sides of Each triangle is 6cm.

Therefore,

→ Area of circle = πr²

→ Area of circle= π(6)²

→ Area of circle = 3.14 * 36

→ Area of circle = 113.04cm².

And,

→ Area of Hexagon = 6 * (Area of Each Equaliteral Triangles with side 6cm.)

→ Area of Hexagon = 6 * {(√3/4) * (6)²}

→ Area of Hexagon = 6 * (36√3/4)

→ Area of Hexagon = 6 * 9√3

→ Area of Hexagon = 54 * 1.73

→ Area of Hexagon = 93.42 cm².

Hence,

→ Shaded Area = Area of circle - Area of Hexagon

→ Shaded Area = 113.04 - 93.42

→ Shaded Area = 19.62 cm². (Ans.)