Is the region contained inside the circles symmetric across the line that connect the intersection points of the circles? If yes, find the area of half of it.
( 200 π3-50sqrt(3) ) square units
Practice makes perfect
Let's draw the circles T and U. Since both circles have the same radius and TU=10, the radius of one circle touches the center of the other circle.
We are asked to find the area of the region contained inside both circles, the shaded region. To find this area, we need to draw several segments as shown below so that we can determine the measure of PUR.
We have created two equilateral triangles inside the shaded region, △ TUP and △ TUR. This means that the measure of ∠ PTR is 120^(∘), and so is the measure of PUR. Note that the shaded region is symmetric across PR, and the half of the area of the shaded region equals the area of the sector determined by PUR.
Finally, we will multiply the area A of the segment by 2 to find the total area of the region contained inside both circles.
2 * A= 2(100 π/3-25sqrt(3) )
⇓
2A = 200 π/3-50sqrt(3)
The region contained inside both circles has an area of 200 π3-50sqrt(3) square units.
