Sue Pemberton Solutions for Chapter: Circular Measure, Exercise 4: EXERCISE 4C

The diagram shows a semicircle, centre $O$, with radius $8 \mathrm{cm}$. $FH$ is the arc of a circle, centre $E$. Find the area of sector $FOG$.

The diagram shows a sector, $EOG$, of a circle, centre $O$, with radius $r \mathrm{cm}$. The line $GF$ is a tangent to the circle at $G$, and $E$ is the midpoint of $OF$. The area of the shaded region is $A {\mathrm{cm}}^2$. Show that $A=\frac{r^2}{6}\left(3\sqrt{3}-\mathrm{\pi }\right)$.

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 $\frac{\mathrm{area} \mathrm{of} \mathrm{inner} \mathrm{circle}}{\mathrm{area} \mathrm{of} \mathrm{sector}}$.