# Area of segments - circle help

• Nov 20th 2009, 09:56 AM
mattyarnold
Area of segments - circle help
I need some help with a puzzle that has been really bugging me. I can't think of any solutions without getting more information, and I have no idea how to get that information.

You have two identical circles that overlap (like a venn diagram) so that the center point of one circle is on the edge of the other circle. If the overlapping area is exactly 20000 square centimetres, what is the radius of the circles?

Either answers or suggestions on how to reach an answer would be great. I know the information is vague... Hence the fact I can't do it ^^
• Nov 20th 2009, 10:08 AM
Plato
• Nov 21st 2009, 07:14 AM
aidan
Quote:

Originally Posted by mattyarnold
I need some help with a puzzle that has been really bugging me. I can't think of any solutions without getting more information, and I have no idea how to get that information.

You have two identical circles that overlap (like a venn diagram) so that the center point of one circle is on the edge of the other circle. If the overlapping area is exactly 20000 square centimetres, what is the radius of the circles?

Either answers or suggestions on how to reach an answer would be great. I know the information is vague... Hence the fact I can't do it ^^

Make a sketch.
.
It should be obvious that the area is EACH segment is half of 20000 or 10,000.

Since "the center point of one circle is on the edge of the other circle",
the circles will intersect at a 60 degree angles from the center points line.
You can prove this by the fact that the Radii are equal, so you have an equilateral triangle.
The overlap of the circles will be an arc of 120 degrees.
The sector-area of the arc of one circle is 1/3 the area of the entire circle: $\dfrac{r^2\pi}{3}$

From that you need to subtract the area of an equilateral triangle with side length of r.
The area of the equilateral triangle is $\dfrac{r^2\sin(60deg)}{2}$.

The area of one segment is 10000.

therefore:

$\dfrac{r^2\pi}{3}$ - $\dfrac{r^2\sin(60deg)}{2}$ = 10000

With a little be of work you can isolate $r^2$ and extract the square root.

.
multiply both side by 6
$2r^2\pi - 3r^2\sin(60deg) = 60000$

simplify for $r^2$
$r^2\left(2\pi - 3\sin(60deg)\right) = 60000$

&divide

$r^2 = \dfrac{60000}{ 2\pi - 3\sin(60deg)}$

$r =\sqrt{ \dfrac{60000}{ 2\pi - 3\sin(60deg)} }$