Hello, a4swe!

Two circles, both with the radius of R (and in the same plane) intersect

so that the centre of one circle lies on the circumference of the other circle.

Calculate the area inside both of the circles.

The intersection is a lens-shaped region. Code:

*
* /:::*
* /::::::*
/:::::::::
* /::::::::::*
/::::::::::::
/::::::::::::::
*/::::::::::::::*
* 120°::::::::::*
*\::::::::::::::*
\::::::::::::::
\::::::::::::
* \::::::::::*
\:::::::::
* \::::::*
*-\:::*
*

In Glaysher's excellent diagram, we see two equilateral triangles.

. . Hence, we have a 120° sector plus two 60° segments.

Since the sector occupies one-third of the circle: .

The area of a segment is: .

. . .

Hence, the area of the *two* segments is: .

Therefore, the area of the intersection is:

. .