Find the area

**seit** · Nov 13th 2010, 10:59 PM

Find the area of the crescent-shaped region(called a lune) bounded by arcs of circles with radii r and R(see the figure)

thank you very much

**HallsofIvy** · Nov 14th 2010, 05:48 AM

There is not enough information. You also need to know the distance between centers of the two circles.

**chisigma** · Nov 14th 2010, 07:50 AM

If R and r are known the distance d between the centers is...

$\displaystyle d=R\ \cos \sin^{-1} \frac{r}{R}$ (1)

Kind regards

$\chi$ $\sigma$

**HallsofIvy** · Nov 14th 2010, 10:11 AM

Oh, yes! I missed the little "right angle" symbol on the r radius!

**Opalg** · Nov 14th 2010, 11:46 AM

The area of the lune is the area of the upper half of the small circle, plus the area of the triangle formed by the two radii of the large circle and the diameter of the small circle (see the attachment), minus the area of the sector of the large circle bounded by those two radii. So the area of the lune is

$\boxed{\tfrac12\pi r^2 + r\sqrt{R^2-r^2} - R^2\arcsin(r/R)}$ .

(Seeing this problem in the Calculus section, I started trying to calculate the area as a double integral using polar coordinates. It's possible to do it that way, but the geometric approach is far easier!)

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Find the area A of the crescent-shaped shaded region (called a lune) bounded by arcs of circles with radii r and R. (See the figure below. Assume the center of the smaller circle is collinear with the two points of intersection.)Find the area A of the cre

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