Question

In figure, arcs have been drawn of radius 21 cm each with vertices A, B, C and D of quadrilateral ABCD as centres. Find the area of the shaded region.

Answer 1

Let r be the radius of each sector = 21 cm

Area of the shaded region = Area of the four sectors

Let angles subtended at A, B, C and D be x°, y°, z° and w° respectively.

Angle subtended at A, B, C, D (in radians, (θ)) be `(xpi)/180, (ypi)/180, (zpi)/180, (wpi)/180` respectively.

∴ Area of a sector with central angle at A = `1/2 r^2 theta`

= `1/2 xx (21)^2 xx (xpi)1/0 = (441xpi)/360  cm^2`

∴ Area of a sector with central angle at B  = `1/2 r^2 theta`

= `1/2 xx (21)^2 xx (ypi)/180 = (441ypi)/360  cm^2`

 Area of a sector with central angle at C = `1/2 r^2 theta`

= `1/2 xx (21)^2 xx (zoi)/180 = (441zpi)/360  cm^2`

∴ Area of a sector with central angle at D = `1/2 r^2 theta`

= `1/2 xx (21)^2 xx (wpi)/180 = (441wpi)/360  cm^2`

∴ Area of four sectors = `((441xpi)/360 + (441ypi)/360 + (441zpi)/360 + (441wpi)/360)  cm^2`

Since, sum of all interior angles in any quadrilateral is 360°

∴ x + y + z +w = 360°

Thus, Area of four sectors = `((441pi)/360) (x + y + z + w)`

= `(441pi)/360 xx 360`

= 441π cm²

= 1386 cm²

Hence, required area of the shaded region is 1386 cm².