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In the figure given below ABCDEF is a regular hexagon and P is the mid-point of AB. If area of ∆PED is 98√3 cm2, then what is the area of shaded region?

In the figure given below ABCDEF is a regular hexagon and P is the mid-point of AB. If area of ∆PED is 98√3 cm2, then what is the area of shaded region? Correct Answer 14(14√3 – 11) cm<sup>2</sup>

Given:

ABCDEF is a regular hexagon

Area of ∆PED is 98√3 cm2

Formula used:

Area of triangle = 1/2 × base × height

Area of regular hexagon = 3√3/2 (side)2

Calculation:

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Let side of hexagon be a

We know that ∆AOB is an equilateral triangle where ‘O’ is the center of hexagon.

Area of ∆AOB = Area of ∆AOF

⇒ (√3/4) × a2 = 1/2 × a × (AE/2)

⇒ AE = a√3

Area of ∆PED = 1/2 × ED × AE

⇒ 98√3 = 1/2 × a × (a√3)

a2 = 196

a = 14 cm

Area of hexagon = 3√3/2 × (a)2

⇒ Area of hexagon = 3√3/2 × 196

⇒ Area of hexagon = 294√3

Now,

Area of semi-circle AF + Area of semi-circle BC = Area of circle with diameter 14 cm.

⇒ Area of semi-circle AF + Area of semi-circle BC = π(7)2 = 154 cm2

Area of shaded region = Area of hexagon – Area of two semi-circle – Area of ∆PED

⇒ Area of shaded region = 294√3 – 154 – 98√3

⇒ Area of shaded region = 196√3 – 154

∴ Area of shaded region = 14(14√3 – 11) cm2

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