Help please, Math question !

Question

What is the area of the shaded segment shown in  below?

Answer 1

The area of the sector is 30π(12)²/360 = 12²π/12 = 12π= 37.699 in² The area of the triangle MNO is found by calculating the height using trig where angle MNO is 75° and so height of triangle is found using the

sin75° = height/12 so height = 12 sin 75°= 11.59 in and the base of the triangle is found using the cos 75°=

x/12 where x is the line segment PN. and so x= 12 cos 75° = 3.106 and the base of the triangle is 6.21 in and so the area of the triangle is 6.21* 11.59/2 = 35.99 in² with this the area of the shaded region is 37.699 -35.99 in² which gives 1.709 in² _{as the area of the shaded region.}

Answer 2

Alternate Solution..... the area of the shaded section is the area of the arc swept area minus the area of the triangle OMN.

The area total circle is pi*r^2 = 144*pi, So the area of the arc swept area is (30/360)*144*pi = 12pi

The area of the triangle OMN is obtained by finding length MN which is gotten by the law of cosines. That result is MN = 6.2117 and the triangle area is found to be 36.

So the area of the shaded area is 12pi - 36 = 1.7 in^2

You can compare this answer with the answer given above but I believe the 30 degree angle in the above answer should have been converted to radians. If so, his solution matches this one!

Answer 3

A = r²(θ - sin θ) / 2

A = 12²(30 - 0.5) / 2

A = 144(2.5) / 2

A = 12(2.5)

A = 30