Question

In Fig., ABCD is a trapezium with AB || DC, AB = 18 cm, DC = 32 cm and distance between AB and DC = 14 cm. If arcs of equal radii 7 cm with centres A, B, C and D have been drawn, then find the area of the shaded region of the figure.

Answer 1

AB = 18 cm, DC = 32 cm

Distance between AB and DC = Height = 14 cm

Now, Area of the trapezium = (1/2) × (Sum of parallel sides) × Height

= (½) × (18+32) × 14 = 350 cm²

As AB ∥ DC, ∴ ∠ A +∠ D = 180°

And ∠ B +∠ C = 180°

Also, radius of each arc = 7 cm

Therefore,

Area of the sector with central angle A = (½) × (∠A/180) × π × r²

Area of the sector with central angle D = (½) × (∠D/180) × π × r²

Area of the sector with central angle B = (½) × (∠B/180) × π × r²

Area of the sector with central angle C = (½) × (∠C/180) × π × r²

Total area of the sectors,

= 77 + 77

= 154

∴ Area of shaded region = Area of trapezium – (Total area of sectors)

= 350 – 154 = 196 cm²

Hence, the required area of shaded region is 196 cm².