Question

find the area of shaded region where circular arc of r is 7cm
has been drawn with vertex a of an equilateral triangle abc of side 14cm as centre (use π=22/7and √3=1.73)

Answer 1

radius of circle is r=7 cm.

So it's are is πr²

=22/7×7²

=22×7

=154cm²

Now ,unshaded area of circle that is (60/360)πr² (the equilateral triangle has angle 60° ) has to be remove from it.

πr²/6

=154/6

=25.67cm²

So total shaded area in circular part is 154 - 25.67=128.33cm²

Now area of equilateral triangle =√3a²/4 where a is the side of triangle =14cm.

√3(196)/4

=49√3

=49×1.73

=84.77cm²

As this area of triangle also contains unshaded area of segment we have to substract from it also.

84.77 - 25.67=59.10cm²

So total shaded area

=59.10+128.33

=187.43cm²

Answer 2

Length of the arc drawn with vertex a of an equilateral triangle = 7 cm

Angle formed by equilateral triangle  = 60º

Find the radius:

Length of arc = θ/360 x 2πr

60/360 x 2πr = 7 cm

1/3 πr = 7

r = 21/π cm

Find the area of the unshaded region:

Area of the unshaded region = θ/360 x πr²

Area of the unshaded region = 60/360 x π(21/π)²

Area of the unshaded region = 73.5/π cm²

Find the area of the circle:

Area = πr²

Area = π (21/π) ²

Area = 441/π cm²

Find the area of the equilateral triangle:

Area of equilateral triangle = √3/4 (side)²

Area of equilateral triangle = √3/4 (14)² = 84.77 cm²

Find the area of the shaded region:

Area = 84.77 + 441/π - 73.5/π - 73.5/π = 178.32 cm²

Answer:  178.32 cm²