finding the shaded area of a triangle

**luckystar001** · August 13th 2010, 10:19 AM

i need help with this problem. i have absolutely no clue how to solve it.

Consider the shaded region outside the sector of the circle of radius 8 meters and inside the right triangle. write the area A of the region as a function of Θ.

Please and Thank you!

**Archie Meade** · August 13th 2010, 10:29 AM

Originally Posted by luckystar001

i need help with this problem. i have absolutely no clue how to solve it.

Consider the shaded region outside the sector of the circle of radius 8 meters and inside the right triangle. write the area A of the region as a function of Θ.

Please and Thank you!

Hi luckystar001,

the shaded region is the difference between the triangle area and a sector of a circle.
Hence the shaded region is

$triangle\ area-sector\ area.$

The triangle area is $0.5(base)(height)$

but since $tan\theta=\frac{height}{8}$

$\Rightarrow\ h=8tan\theta\ \Rightarrow\ triangle\ area=0.5(8)(8tan\theta)$

The area of the complete circle is ${\pi}r^2={\pi}8^2$

The area of the sector is a fraction of the area of the full circle.

That fraction is $\frac{\theta}{360^o}$

Can you complete?

**luckystar001** · August 13th 2010, 10:54 AM

Thank you very much! I wouldn't have thought of that. You're a genius!

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