Thread: Area - circles and equilateral triangle.

1. Area - circles and equilateral triangle.

1. Find the area of the shaded region. this triangle is equilateral all sides are 2.

Thanks for helping!

2. 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.