In the given figure, the center of the circle is A and ABCDEF is a regular hexagon of side 6 cm. What is the approximate area of segment BPF?

Question

In the given figure, the centre of the circle is A and ABCDEF is a regular hexagon of side 6 cm. What is the approximate area of segment BPF? (Take π = 3.14)

(a) 25 cm² 

(b) 22 cm² 

(c) 32 cm² 

(d) 30 cm²

Answer 1

Answer : (b) 22 cm² 

 ∠BAF is an angle of the regular hexagon, ∠BAF = 120º.

Draw AK ⊥ BF. The perpendicular from the center of the circle to a chord bisects the chord, so BK = KF. 

Also ∆ABF being an isosceles triangle, ∠BAK = \(\frac{1}{2}\) ×∠BAF = 60º 

∴ ∠ABK = 30º 

Also radius of the circle = side AB of hexagon = 6 cm.

In ∆ABK, AK = AB sin 30º = 6 × \(\frac{1}{2}\) = 3 cm, and 

BK = AB cos 30º = 6 × \(\frac{√3}{2}\) = 3√3 cm

∴ BF = 2 × BK = 6√3 cm.

Now, area of segment BPF = Area of sector ABPF – Area of △ABF

= \(\frac{120}{360}\) × 3.14 × 6² - \(\frac{1}{2}\) × 6√3 × 3 

= 37.68 cm² - 15.57 cm²

= 22.11 cm² ~ 22 cm²