# 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 are - Mathematics

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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.

#### Solution

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.

Concept: Areas of Sector and Segment of a Circle
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#### APPEARS IN

NCERT Mathematics Exemplar Class 10
Chapter 11 Area Related To Circles
Exercise 11.4 | Q 6 | Page 133
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