The diagram shows two circles with radius $r \mathrm{cm}$. The centre of each circle lies on the circumference of the other circle. Find, in terms of $r$, the exact area of the shaded region.

The diagram shows a square of side length $10 \mathrm{cm}$. A quarter circle, of radius $10 \mathrm{cm}$, is drawn from each vertex of the square. Find the exact area of the shaded region.

The diagram shows a circle with radius $1 \mathrm{cm}$, centre $O$. Triangle $AOB$ is right angled and its hypotenuse $AB$ ia a tangent to the circle at $P$. Angle $BAO=x$radians. Find the expression for the length of $AB$ in terms of $\tan x$.

The diagram shows a circle with radius $1 \mathrm{cm}$, centre $O$. Triangle $AOB$ is right angled and its hypotenuse $AB$ ia a tangent to the circle at $P$. Angle $BAO=x$radians. Find the value of $x$ for which the two shaded areas are equal.[Enter the value of x which satisfies $x>1$, round off to two decimal places]

The diagram shows a sector, $AOB$, of a circle, centre $O$, with radius $R \mathrm{cm}$ and sector angle $\frac{\mathrm{\pi }}{3}$ radians. An inner circle of radius $r \mathrm{cm}$ touches the three sides of the sector. Find the value of $R$ in terms of  $r$.

The diagram shows a sector, $AOB$, of a circle, centre $O$, with radius $R \mathrm{cm}$ and sector angle $\frac{\mathrm{\pi }}{3}$ radians. An inner circle of radius $r \mathrm{cm}$ touches the three sides of the sector. Find $\frac{\mathrm{area} \mathrm{of} \mathrm{inner} \mathrm{circle}}{\mathrm{area} \mathrm{of} \mathrm{sector}}$.

The diagram shows an equilateral triangle $PQR$, with side length $5 \mathrm{cm}$. $M$ is the midpoint of the line $QR$. An arc of a circle, centre $P$, touches $QR$ at $M$ and meets $PQ$ at $X$ and $PR$ at $Y$. Find in terms of $\mathrm{\pi }$ and $\sqrt{3}$ : the total perimeter of the region $RYXQ$.

The diagram shows an equilateral triangle $PQR$, with side length $5 \mathrm{cm}$. $M$ is the midpoint of the line $QR$. An arc of a circle, centre $P$, touches $QR$ at $M$ and meets $PQ$ at $X$ and $PR$ at $Y$. Find in terms of $\mathrm{\pi }$ and $\sqrt{3}$  the total area of the shaded region $RYXQ$.

In the diagram, $OAB$ is a sector of a circle with centre $O$ and radius $8 \mathrm{cm}$. Angle $BOA$ is $\alpha$ radians. $OAC$ is a semicircle with diameter $OA$.The area of the semicircle $OAC$ is twice the area of the sector $OAB$.Find $\alpha$ in terms of $\pi$.