Question

Find the area of the shaded portion in the circle.


help me click pic below i had got 18pi - 9 sqrt3

Answer 1

Area = the semi-circle area + the area of the segment.\[\frac{\pi r^2}{2}+\frac{ r^2}{2}\left(\left(\pi -2\text{ArcSin}\left[\frac{h}{r}\right]\right)-\left(\text{Sin}\left[\pi -2\text{ArcSin}\left[\frac{h}{r}\right]\right]\right)\right) \]where h is the height of the segment above the diameter, "3" in the attachment, and r is the circle's radius. Using Mathematica, the result of calculating the expression limit as h approaches zero is shown below:\[\text{Limit}\left[\frac{\pi r^2}{2}+\frac{ r^2}{2}\left(\left(\pi -2 \text{ArcTan}\left[\frac{h}{r}\right]\right)-\text{Sin}\left[\pi -2 \text{ArcTan}\left[\frac{h}{r}\right]\right]\right),h\to 0\right] = \pi r^2 \]Assume that the circle's radius is 1/2 of 12, and that the height is 3, then, the area in question is:\[30 \pi -9 \sqrt{3}=78.6593 \]For a radius of 12 and an height of 3 the area is:\[-9 \left(\sqrt{15}-16 \pi +16 \text{ArcCsc}[4]\right)=381.147\]

Answer 2

Thanks for the medal.