: How to find the radius of two identical circles?
So, I have a math problem where I need to find the Radius of two identical, overlapping circles. They give me the area of the overlapping area (194 sq feet). Also, the center of one circle is on the overlapping line of the other one… anyone know how to help me? I have looked all over the internet, and I need it for tomorrow!
Make sure to make and label a sketch (if you haven’t already)
If the circles overlap each other’s center, that means they are a radius apart. A line from the center to a point of intersection will also be a radius long (and since you can draw it from either center, that means that you can create an equilateral triangle between the two centers and a point of intersection.
The included angle will be 60 degrees. You can then do the same to the other point of intersection, meaning the angle from one point of intersection to one center to the other point of intersection is 2(60) = 120 degrees
This will then cut a sector that is 1/3 (120 / 360) of the entire area
This sector, pi * r^2 / 3, will consist of two equilateral triangles of side r (and area sqrt(3) / 4 * r^2)) and a “sliver” between the triangle and the circumference.
The entire overlapping area will contain two of these slivers.
Thus: area of two triangles = 2r^2 (sqrt(3) / 4) = r^2 sqrt(3) / 2
area of each sliver: pi * r^2 / 3 – r^2 sqrt(3) / 2 = r^2 [pi / 3 - sqrt(3) / 2]
overlapping area = r^2 (sqrt(3) / 2) + 2r^2 [pi/3 - sqrt(3) / 2] = 194
r^2 [sqrt(3) / 2 + 2pi/3 - sqrt(3)] = 194
r^2 [2pi/3 - sqrt(3) / 2] = 194
r^2 = 194 / [2pi/3 - sqrt(3) / 2]
thus, r = sqrt[194 / [2pi/3 - sqrt(3) / 2]
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