Area - circles and equilateral triangle.

• Oct 14th 2010, 12:20 AM
ejaykasai
Area - circles and equilateral triangle.
1. Find the area of the shaded region. this triangle is equilateral all sides are 2.
http://i1209.photobucket.com/albums/.../Photo0238.jpg

Thanks for helping! :)
• Oct 14th 2010, 03:38 AM
HallsofIvy
You have an equilateral triangle with sides of length 2. If you drop a perpendicular from one vertex of the triangle to the opposite side, you divide the triangle into two right triangles with hypotenuse 2 and one leg 1. You can use the Pythagorean theorem to determine that $\displaystyle x^2+ 1^2= 2^2$, $\displaystyle x^2= 4- 1= 3$, $\displaystyle x= \sqrt{3}$: the other leg has length $\displaystyle \sqrt{3}$, the altitude of the triangle. From that you can calculate the area of the triangle.

Each of the three circles has radius 1 and so area $\displaystyle \pi$. Each angle of the triangle is 60 degrees which is 60/360= 1/6 of the complete circle. That is, each of the sectors of a circle has area $\displaystyle \frac{\pi}{6}$.

The shaded area is the area of triangle minus the three sector areas.