geometry question on areas in arcs PS is a line segment of length 4 and O is the midpoint of PS. A semicircular arc is drawn with PS as diameter. Let X be the midpoint of this arc. Q and R are points on the arc PXS such that QR is parallel to PS and the semicircular arc drawn with QR as diameter is tangent to PS. How can I get the area of the region QXROQ bounded by the two semicircular arcs?
 A: This diagram will help:

Note that I have connected $OQ$ and $OR$ - these are both equal to the radius of the blue circle centered at $O$ and so their lengths must be equal to $2$.
I have also labelled the intersection of $QR$ and $OX$ as $Y$. From the question statement it is clear that:$$QY=YR=OY$$
and I have labelled this length as $a$.
Now note in triangle $OYR$ that $OY=YR$ and angle $OYR=90^{\circ}$. This implies that angles $YOR=YRO=45^{\circ}$.
Using similar arguments you can deduce that angles $YOQ=YQO=45^{\circ}$.
This should give you enough information to be able to calculate the value of $a$ and hence calculate the area you require.
A: Let $S$ be the intersection point between $OX$ and $QR$. Also, let $QS=r$. 
I think you can get the area you want using the followings :


*

*$QS=SR=SO=r$.

*$\triangle{SQO}$ and $\triangle{SRO}$ are both isosceles right triangles.

*$\angle{QOR}=\angle{SOQ}+\angle{SOR}=\frac{\pi}{4}+\frac{\pi}{4}=\frac{\pi}{2}$.

*The area we want is given by $[\text{semicircle $QRO$}]+[\text{figure $QXR$}]$.

*$[\text{figure $QXR$}]=[\text{sector $OQXR$}]-[\triangle{OQR}]$.
