Question

In figure, 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

= `1/2` × 18 + 32 × 14 = 350cm²

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 = `1/2 xx (∠A)/180` × π × r²

Area of the sector with central angle D = `1/2 xx (∠D)/180` × π × r²

Area of the sector with central angle B = `1/2 xx (∠B)/180` × π × r²

Area of the sector with central angle C = `1/2 xx (∠C)/180` × π × r²

Total area of the sectors,

= `(∠A)/360 xx pi xx r^2 + (∠D)/360 xx pi xx r^2 + (∠B)/360 xx pi xx r^2 + (∠C)/360 xx pi xx r^2`

= `((∠A + ∠D)/360 xx pi xx r^2) + ((∠B + ∠C)/360 xx pi xx r^2)`

= `(180/360 xx 22/7 xx 49) + (180/360 xx 22/7 xx 49)`

= 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.