How do I find the area of the shaded part of the circle?

Question

anonymous · Answer

attachment

campbell_st · Answer

lol... I did this a while ago

its area of a kite - area of the sector

the sector is the easy part

$$A = \frac{120}{360}\pi 6^2$$

$$A = 12\pi$$

next find the diagonals of the kite

the vertical diagonal (chord) 

|dw:1339794554221:dw|

use the cosine rule

$$c = 6^2 + 6^2 - 2\times6\times6\times \cos(120)$$

$$c = 6\sqrt{3}$$

the secant length can be found using tri

|dw:1339794674882:dw|

$$\cos(60) = \frac{6}{x}$$

x = 12

Area of the kite 

$$A = \frac{1}{2}\times12\times6\sqrt{3}$$

$$A = 36\sqrt{3}$$

$$shaded area  = 36\sqrt{3} - 12\pi$$

or $$Area = 12(3\sqrt{3} - \pi)$$