Circular Measure (Radians) (Cambridge (CIE) A Level Maths: Pure 1): Exam Questions

Complete the table.

A sector of a circle $OPQ$ has radius 3.4 cm, and the angle at the centre $O$ is $\frac{3\pi }{4}$ radians.

(i) Find the length of the arc $PQ$.

(ii) Find the area of the sector $OPQ$.

The diagram shows a sector of a circle with centre $O$ and radius 9 cm. The angle at the centre is $\frac{\pi }{5}$ radians.

Find the area and the perimeter of the sector.

A rainbow-shaped logo is formed from two semicircles, as shown below. The radius of the smaller semicircle is 1 cm less than the radius of the larger semicircle.

Find the area of the logo.

The diagram shows a sector of a circle $OAB$ with centre $O$. The radii $OA$ and $OB$ are each 5.4 cm, and the angle $AOB$ is 1.2 radians.

Giving each answer correct to 3 significant figures:

(i) find the area of the sector $OAB$;

(ii) find the area of the triangle $OAB$;

(iii) find the area of the shaded segment.

(i) Find the length of the arc $AB$.

(ii) Find the perimeter of the sector $OAB$.

The canopy of a parachute and the two outermost connecting cords form a sector of a circle, as shown below, with the parachutist modelled as a particle at the point $O$. The angle at the centre $O$ is $\frac{\pi }{4}$ radians.

The area of the sector $OAB$ is $\frac{81\pi }{200}$ m².

Find the length of one of the connecting cords.

A plastic puzzle piece is a prism whose cross-section is a sector of a circle, as shown below. The sector has radius 8 cm and the angle at the centre is 1.2 radians. The prism is 2 cm thick.

(i) Work out the area of the cross-section.

(ii) Hence, or otherwise, work out the volume of the puzzle piece.

The diagram shows a circle sector $OAB$ with centre $O$. The angle at the centre is $\frac{\pi }{3}$ radians, and the line segments $OC$ and $BC$ have lengths 8 cm and $p$ cm respectively. Also, $CD$ is parallel to $AB$, so that $AD=BC$ and $OD=OC$.

Show that the area of the sector $OAB$ is $\frac{\pi }{6}(p+8{)}^2$ cm².

Show that the area of the triangle $OCD$ is $16\sqrt{3}$ cm².

Given that the area of the shaded shape $ABCD$ is $\left(\frac{50\pi }{3}-16\sqrt{3}\right)$ cm², find the value of $p$.

A sector of a circle has area $8\pi$ cm² and arc length $4\pi$ cm.

Find the radius of the circle and the angle at the centre of the sector.

The canopy of a parachute and the two outermost suspension lines form a sector of a circle, as shown below, with the parachutist modelled as a particle at the point $O$. The angle at the centre $O$ is $\frac{\pi }{3}$ radians.

The area of the sector $OAB$ is $\frac{49\pi }{150}$ m².

The parachute is made with 40 equal-length suspension lines. Disregarding any extra cord needed for knots and fastenings, find the total length of suspension line required to make the parachute.

A sector of a circle $OPQ$ has radius $a$ cm, and the angle at the centre $O$ is $\frac{a}{3}$ radians.

Given that the area of the sector $OPQ$ (in cm²) is three times the length of the arc $PQ$ (in cm), find the value of $a$.

The diagram shows a sector of a circle $OAB$ with centre $O$ and radius $r$ cm. The angle at the centre is 0.8 radians.

Given that the area of the triangle $OAB$ is 5.64 cm², find the area of the shaded segment, giving your answer correct to 3 significant figures.

Find the perimeter of the sector $OAB$, giving your answer correct to 3 significant figures.

The diagram shows a prism whose cross-section is a sector of a circle. The sector has radius $(2a-3)$ cm and the angle at the centre is 1.7 radians. The prism is 4 cm thick.

Given that the volume of the prism is 7.65 cm³, find the possible value(s) of $a$.

A rainbow-shaped logo is formed from three semicircles, as shown below. Each inner semicircle has a radius that is 1 cm less than that of the next semicircle further out.

Find, in its simplest form, the ratio of the outer area $A$ of the logo to the inner area $B$.

The canopy of a parachute and the two outermost suspension lines form a sector of a circle $OAB$, as shown below, with the parachutist modelled as a particle at the point $O$.

The area of the sector $OAB$ is $\frac{125\pi }{144}$ m², and the length of the arc $AB$ is $\frac{25\pi }{36}$ m.

Find the length of one suspension line, and the angle $AOB$ that the two outermost suspension lines make at the parachutist.

The diagram shows a sector of a circle $OAB$ with centre $O$, radius $r$ cm and angle $\theta$ radians at the centre.

Show that the area of the shaded segment is $\frac{1}{2}{r}^2(\theta -\text{sin} \theta )$ cm².

Find, in terms of $\theta$, the percentage of the sector that the segment occupies.

An eccentric wizard has captured a unicorn and will only set it free if you can answer the following question:

"I wish to create a rainbow-shaped mosaic for the floor of the throne room in my castle. The mosaic is to be formed from four semicircles, as shown below. The innermost semicircle has radius $R$ metres, and each outer semicircle has a radius that is a constant $k$ metres greater than the one immediately inside it. The rainbow mosaic must be exactly 22 metres across.

I also wish the inner region (labelled $A$ in the diagram) to take up exactly one quarter of the total area of the rainbow."

Find the required values of $R$ and $k$, and set the unicorn free.

A sector of a circle $OST$ has radius $r$ cm, and the angle at the centre $O$ is $\theta$ radians. The chord $ST$ has length $a$ cm.

Show that $a^2=2r^2(1-\text{cos} \theta )$.

Given that $r=4\theta$ and the area of the sector is $\frac{8{\pi }^3}{27}$ cm², find the value of $a$.

The diagram shows a sector of a circle with centre $O$. The radii $OA$ and $OB$ are each $r$ cm, and the angle at the centre $AOB$ is $\theta$ radians. The point $D$ lies on $OA$ and the point $C$ lies on $OB$, with $DC$ perpendicular to $OB$.

Given that $BC:CO=2:3$, show that the area of the shaded shape $ABCD$ is $\frac{1}{50}{r}^2(25\theta -9 \text{tan} \theta )$ cm².