Circular Measure (Cambridge (CIE) O Level Additional Maths): Exam Questions

The diagram shows a rectangle $ABCD$ and an arc $AXB$ of a circle with centre at $O$, the midpoint of $DC$.
The length of $BC$ is $8\mathrm{cm}$ and the length of $DC$ is $30\mathrm{cm}$.
Angle $AOB$ is $\theta$ radians.

Find the perimeter of the shape $ADOCBX$.

Find the area of the shape $ADOCBX$.

In this question all lengths are in centimetres.

The diagram shows a shaded shape. The arc $AB$ is the major arc of a circle, centre $O$, radius 10. The line $AB$ is of length 15, the line $OC$ is of length 25 and the lengths of $AC$ and $BC$ are equal.

Show that the angle $AOB$ is 1.70 radians correct to 2 decimal places.

Find the perimeter of the shaded shape.

Find the area of the shaded shape.

In this question all lengths are in centimetres and all angles are in radians.

The diagram shows the rectangle $ADEF$, where $AF=DE=r$. The points $B$ and $C$ lie on $AD$ such that $AB=CD=r$ . The curve $BC$ is an arc of the circle, centre $O$, radius $r$ and has a length of $1.5r$.

Show that the perimeter of the shaded region is $(7.5+2\sin 0.75)r$ .

Find the area of the shaded region, giving your answer in the form $k{r}^2$, where $k$ is a constant correct to 2 decimal places.

The diagram shows an isosceles triangle $OAB$ such that $OA=OB$ and angle $AOB=\theta$ radians. The points $C$ and $D$ lie on $OA$ and $OB$ respectively. $CD$ is an arc of length 9.6 cm of the circle, centre $O$, radius 12 cm. The arc $CD$ touches the line $AB$ at the point $M$.

Find the value of $\theta$.

Find the total area of the shaded regions.

Find the total perimeter of the shaded regions.

The circles with centres $C_1$ and $C_2$ have equal radii of length $r\mathrm{cm}$. The line $C_1{C}_2$ is a radius of both circles. The two circles intersect at $A$ and $B$.

Given that the perimeter of the shaded region is $4\mathrm{\pi }\mathrm{cm}$, find the value of $r$. [4]

Find the exact area of the shaded region.

In this question all lengths are in centimetres.

The diagram shows the figure $ABC$. The arc $AB$ is part of a circle, centre $O$, radius $r$, and is of length $1.45r$. The point $O$ lies on the straight line $CB$ such that $CO=0.5r$.

Find, in radians, the angle $AOB$.

Find the area of $ABC$, giving your  in the form $k{r}^2$, where $k$ is a constant.

Given that the perimeter of $ABC$ is $12\mathrm{cm}$, find the value of $r$.

The diagram shows a shape consisting of two circles of radius 3 cm and 4 cm with centres $A$ and $B$ which are 5 cm apart. The circles intersect at $C$ and $D$ as shown. The lines $AC$ and $BC$ are tangents to the circles, centres $B$ and $A$ respectively. Find the angle $CAB$ in radians.

Find the perimeter of the whole shape.

Find the area of the whole shape.