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 xx "Sum of parallel sides" xx "Height"`

= `1/2 xx 18 + 32 xx 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 = `1/2 xx (∠"A")/180^circ xx π xx "r"^2`

Area of the sector with central angle D = `1/2 xx (∠"D")/180^circ xx π xx "r"^2`

Area of the sector with central angle B = `1/2 xx (∠"B")/180^circ xx π xx "r"^2`

Area of the sector with central angle C = `1/2 xx (∠"C")/180^circ xx π xx "r"^2`

Total area of the sectors,

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

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

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