# finding the shaded area of a triangle

• Aug 13th 2010, 10:19 AM
luckystar001
finding the shaded area of a triangle
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.

• Aug 13th 2010, 10:29 AM
Quote:

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.

Hi luckystar001,

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

$\displaystyle triangle\ area-sector\ area.$

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

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

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

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

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

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

Can you complete?
• Aug 13th 2010, 10:54 AM
luckystar001
Thank you very much! I wouldn't have thought of that. You're a genius!