Circular measure

In Fig. $1$, $OAB$ is a sector of a circle with centre $O$ and radius $r$. $AX$ is tangent to the arc $AB$ at $A$, and $\angle BAX = \alpha$. Show that $\angle AOB = 2\alpha$.

Feb/March 2016

The diagram indicates that $AB = AC = 8$ cm, while angle $CAB = \frac{2}{7}\pi$ radians. Circular arc $BC$ is centred at $A$, circular arc $CD$ is centred at $B$, and $ABD$ lies on one straight line.

Feb/March 2017

The diagram depicts sector $ABC$, which is a section of a circle with radius $a$. Points $D$ and $E$ are located on $AB$ and $AC$ respectively, with $AD = AE = ka$, where $k < 1$. Line $DE$ splits the sector into two areas of equal size.

Feb/March 2021

The diagram illustrates a metal plate $ABCDEF$ formed by removing the two shaded regions from a circle of radius $10$ cm and centre $O$. The parallel sides $AB$ and $ED$ each have length $12$ cm.

May/June 2010

In the diagram, $OAB$ is an isosceles triangle with $OA = OB$ and angle $AOB = 2\theta$ radians. Arc $PST$ has centre $O$ and radius $r$, and line $ASB$ is tangent to arc $PST$ at $S$.

May/June 2011

In the diagram, $AB$ forms an arc of a circle with centre $O$ and radius $6$ cm, and $AOB = \frac{\pi}{3}$ radians. The line $AX$ touches the circle at $A$, and $OBX$ lies on a straight line.

May/June 2011

The diagram shows an equilateral triangle $ABC$ with side $2\text{ cm}$. Point $Q$ lies at the midpoint of $BC$. A circular arc with centre $A$ is tangent to $BC$ at $Q$, and it intersects $AB$ at $P$ and $AC$ at $R$.

May/June 2012

In the figure, $AB$ is a circular arc with centre $O$ and radius $r$. The segment $XB$ touches the circle at $B$, and $A$ is the midpoint of $OX$.

May/June 2012

The diagram shows that $OAB$ is a sector of a circle with centre $O$ and radius $8\text{ cm}$. The angle $BOA$ equals $\alpha$ radians. $OAC$ is a semicircle whose diameter is $OA$. The area of the semicircle $OAC$ is twice the area of the sector $OAB$.

May/June 2013

The diagram depicts a square $ABCD$ with side length $10\text{ cm}$. Point $O$ is the midpoint of $AD$, and $BXC$ is an arc of a circle centred at $O$.

May/June 2013

The diagram illustrates a circle $C$ with centre $O$ and radius $3\,\text{cm}$. The radii $OP$ and $OQ$ are produced to $S$ and $R$ respectively, making $ORS$ a sector of a circle centred at $O$. Given that $PS = 6\,\text{cm}$ and that the shaded region has the same area as circle $C$.

May/June 2013

The diagram represents triangle $ABC$, with $AB$ perpendicular to $BC$. $AB$ measures $4 \text{ cm}$ and angle $CAB$ is $\alpha$ radians. The arc $DE$ has centre $A$ and radius $2 \text{ cm}$, and it cuts $AC$ at $D$ and $AB$ at $E$.

May/June 2014

The diagram represents a sector of a circle with centre O and radius r cm. The chord AB splits the sector into triangle AOB and segment AXB. Angle AOB measures \theta radians.

May/June 2014

The diagram depicts a part of a circle with centre $O$ and radius $6\text{ cm}$. The chord $AB$ is arranged so that angle $AOB = 2.2$ radians.

May/June 2014

The diagram shows $AYB$ as a semicircle with $AB$ as its diameter, while $OAXB$ is a sector of a circle centred at $O$ with radius $r$. The angle $AOB$ is $2\theta$ radians.

May/June 2015

The diagram shows $OAB$ as a sector of a circle centred at $O$ with radius $r$. Point $C$ lies on $OB$ so that the angle $ACO$ is a right angle. The angle $AOB$ is $\alpha$ radians, and $AC$ splits the sector into two parts of equal area.

May/June 2015

In the diagram, $AOB$ makes a quarter circle with centre $O$ and radius $r$. Point $C$ is on arc $AB$, and point $D$ is on $OB$. The line $CD$ runs parallel to $AO$, and angle $AOC = \theta$ radians.

May/June 2016

The figure depicts a circle of radius $r$ cm with centre $O$. $PT$ is tangent to the circle at $P$, and $POT = \alpha$ radians. The line $OT$ intersects the circle at $Q$.

May/June 2016

The figure shows $OAXB$ as a sector of a circle centred at $O$ with radius $10\,\text{cm}$. The chord $AB$ has length $12\,\text{cm}$. Line $OX$ passes through $M$, the midpoint of $AB$, and $OX$ is perpendicular to $AB$. The shaded part is enclosed by the chord $AB$ and the arc of a circle centred at $X$ with radius $XA$.

May/June 2017

The diagram depicts a circle of radius $r$ cm with centre $O$. $A$ and $B$ are points on the circumference, and $ABCD$ is a rectangle. The angle $AOB = 2\theta$ radians, and $AD = r$ cm.

May/June 2017

The diagram displays two circles, one centred at $A$ and the other at $B$, with radii $8\text{ cm}$ and $10\text{ cm}$ respectively. They meet at $C$ and $D$, where $CAD$ is a straight line and $AB$ is perpendicular to $CD$.

May/June 2017

The diagram depicts a circle with centre $O$ and radius $r$ cm. Points $A$ and $B$ are on the circle, and $AT$ is tangent to the circle. Angle $AOB = \theta$ radians, and $OBT$ is a straight line.

May/June 2018

In the diagram, $A$ and $B$ lie on a circle of centre $O$ and radius $r$. The tangents at $A$ and $B$ intersect at $T$. The shaded region is enclosed by the minor arc $AB$ together with the lines $AT$ and $BT$. Angle $AOB$ measures $2\theta$ radians.

May/June 2018

A sector in a circle with radius $r$ cm has area $A$ cm$^2$. Write down the perimeter of the sector in terms of $r$ and $A$.

May/June 2019

The diagram shows $OAB$ as a sector of a circle with centre $O$ and radius $2r$, with angle $AOB = \frac{\pi}{6}$ radians. Point $C$ lies halfway along $OA$.

May/June 2020

The diagram depicts a cord looped around a pulley and a pin. The pulley is represented by a circle with centre $O$ and radius $5\text{ cm}$. The thickness of the cord and the dimensions of the pin $P$ may be ignored. The pin is placed $13\text{ cm}$ vertically beneath $O$. Points $A$ and $B$ lie on the circle so that $AP$ and $BP$ are tangents to the circle. The cord runs over the major arc $AB$ of the circle and beneath the pin so that the cord stays taut.

May/June 2020

The diagram illustrates a symmetrical metal plate. It is formed by cutting out two congruent parts from a circular disc with centre $C$. The outline of the plate is made up of the two arcs $PS$ and $QR$ from the original circle, together with two semicircles having $PQ$ and $RS$ as diameters. The radius of the circle with centre $C$ is $4\text{ cm}$, and $PQ = RS = 4\text{ cm}$ as well.

May/June 2021

The diagram presents a cross-section of seven cylindrical pipes, each with radius $20\text{ cm}$, secured by a thin rope that is pulled taut around them. The centres of the six outer pipes are $A, B, C, D, E$ and $F$. Points $P$ and $Q$ mark the places where the straight parts of the rope touch the pipe centred at $A$.

May/June 2021

The figure presents triangle $ABC$, where angle $ABC = 90^\circ$ and $AB = 4\text{ cm}$. Sector $ABD$ is part of a circle centred at $A$. Its area is $10\text{ cm}^2$.

May/June 2021

The diagram depicts sector $ABC$ from a circle whose centre is $A$ and whose radius is $r$. The line $BD$ is at right angles to $AC$. The angle $CAB$ is $\theta$ radians.

May/June 2022

The diagram depicts the sector $OBAC$ of a circle that has centre $O$ and radius $10\,\text{cm}$. Point $P$ is on $OC$, and $BP$ is at right angles to $OC$. The angle $AOC = \frac{\pi}{6}$, while the arc length $AB$ is $2\,\text{cm}$.

May/June 2022

In the diagram, triangle $ABC$ has $AB = BC = 6\,\text{cm}$ and $\angle ABC = 1.8$ radians. The arc $CD$ is part of a circle with centre $A$, while $ABD$ is a straight line.

May/June 2022

The diagram depicts sector $OAB$ of a circle whose centre is $O$. The angle $AOB = \theta$ radians, and $OP = AP = x$.

May/June 2023

The figure shows $AOD$ and $BC$ as a pair of parallel straight lines. Arc $AB$ lies on a circle with centre $O$ and radius $15\text{ cm}$. The angle $BOA$ is $\theta$ radians. Arc $CD$ lies on a circle with centre $O$ and radius $10\text{ cm}$. The angle $COD$ is $\frac{1}{2}\pi$ radians.

May/June 2024

The diagram depicts a symmetric plate $ABCDEF$. The line $ABCD$ is straight, and $BC$ measures $2\text{ cm}$. The two sectors $ABF$ and $DCE$ each have radius $r\text{ cm}$, and each angle $ABF$ and $DCE$ is $\frac{1}{3}\pi$ radians.

May/June 2024

The diagram depicts a sector of a circle with centre $C$. The radii $CA$ and $CB$ each measure $r$ cm, and the reflex angle $ACB$ has size $\theta$ radians. The shaded sector has perimeter $65$ cm and area $225\text{ cm}^2$.

May/June 2024

The diagram depicts a sector $ABC$ of a circle with centre $A$ and radius $r$ cm. The angle $BAC$ is $\alpha$ radians, where $0 < \alpha < \frac{1}{2}\pi$.

May/June 2025

The diagram represents the circle with equation $x^2 + y^2 - 14x + 8y + 36 = 0$ together with the line $y = -2$. This line cuts the circle at $A$ and $B$. The circle’s centre is $C$.

May/June 2025

The diagram depicts a square $ABCD$ with each side measuring $12\ \text{cm}$. The points $E$ and $F$ are located on the sides $BC$ and $CD$ respectively, and satisfy $BE = \frac{1}{3}BC$ and $DF = \frac{1}{3}DC$. The arc $EF$ is a segment of a circle centred at $A$. The shaded region is enclosed by the arc $EF$ and the line segments $EC$ and $FC$.

May/June 2025

A sector $ABD$ of a circle is shown, with centre $A$ and radius $10\text{ cm}$. The perpendicular bisector of $AB$ passes through $D$.

May/June 2025

The diagram contains two circles, $C_1$ and $C_2$, which are touching at point $T$. $C_1$ has centre $P$ and radius $8\,\text{cm}$, while $C_2$ has centre $Q$ and radius $2\,\text{cm}$. The points $R$ and $S$ lie on $C_1$ and $C_2$ respectively, and $RS$ is tangent to both circles.

Oct/Nov 2010

The diagram depicts points $A$, $C$, $B$, $P$ on the circumference of a circle whose centre is $O$ and whose radius is $3\ \text{cm}$. Angle $AOC=2.3$ radians.

Oct/Nov 2010

The diagram shows a metal plate $OABC$, made up of a sector $OAB$ of a circle with centre $O$ and radius $r$, along with a triangle $OCB$ that is right-angled at $C$. Angle $AOB = \theta$ radians, and $OC$ is perpendicular to $OA$.

Oct/Nov 2011

The diagram depicts the sector $OAB$ of a circle whose centre is $O$ and whose radius is $r$. The angle $AOB$ is $\theta$ radians. Point $C$ lies on $OA$ so that $BC$ is perpendicular to $OA$. Point $D$ is located on $BC$, and the circular arc $AD$ has centre $C$.

Oct/Nov 2012

The diagram represents a sector of a circle whose centre is $O$ and whose radius is $20\,\text{cm}$. Inside the sector is a circle with centre $C$ and radius $x\,\text{cm}$, touching the sector at $P$, $Q$ and $R$. Angle $POR = 1.2\,\text{radians}$.

Oct/Nov 2012

In the diagram, D is on side AB of triangle ABC, and CD is an arc from a circle centred at A with radius 2\,\text{cm}. BC has length 2\sqrt{3}\,\text{cm} and meets AC at right angles.

Oct/Nov 2012

The diagram illustrates a metal plate formed by fixing together two pieces, $OABCD$ (shaded) and $OAED$ (unshaded). The piece $OABCD$ is a minor sector of a circle with centre $O$ and radius $2r$. The piece $OAED$ is a major sector of a circle with centre $O$ and radius $r$. Angle $AOD$ is $\alpha$ radians. Simplify your answers where possible and find, in terms of $\alpha$, $\pi$ and $r$.

Oct/Nov 2013

Fig. 1 illustrates a hollow cone without a base, constructed from paper. Its radius is $6\text{ cm}$ and its height is $8\text{ cm}$. The paper is sliced from $A$ to $O$ and then unfolded to create the sector shown in Fig. 2. The circular lower edge of the cone in Fig. 1 becomes the arc of the sector in Fig. 2. The sector angle is $\theta$ radians. Calculate

Oct/Nov 2013

The diagram represents sector $OAB$, with centre $O$ and radius $11\,\text{cm}$. The angle $AOB$ is $\alpha$ radians. Points $C$ and $D$ are on $OA$ and $OB$ respectively. Arc $CD$ is centred at $O$ and has radius $5\,\text{cm}$.

Oct/Nov 2013

The diagram shows $AB$ as an arc of a circle centred at $O$ with radius $4\,\text{cm}$. The angle $AOB$ is $\alpha$ radians. Point $D$ lies on $OB$ so that $AD$ is perpendicular to $OB$. The arc $DC$, also centred at $O$, intersects $OA$ at $C$.

Oct/Nov 2014

The diagram depicts triangle $AOB$, where $OA$ measures $12\,\text{cm}$, $OB$ measures $5\,\text{cm}$, and angle $AOB$ is a right angle. Point $P$ lies on $AB$, and $OP$ is an arc of a circle with centre $A$. Point $Q$ lies on $AB$, and $OQ$ is an arc of a circle with centre $B$.

Oct/Nov 2014

The diagram depicts a metal plate $OABC$, made up of a right-angled triangle $OAB$ and a sector $OBC$ from a circle with centre $O$. Angle $AOB = 0.6$ radians, $OA = 6\,\text{cm}$ and $OA$ is perpendicular to $OC$.

Oct/Nov 2015

The diagram depicts a metal plate $OABCDEF$ made up of $3$ sectors, all with centre $O$. Sector $COD$ has radius $2r$ and angle $COD$ is $\theta$ radians. Each of the sectors $BOA$ and $FOE$ has radius $r$, and $AOED$ and $CBOF$ are straight lines.

Oct/Nov 2015

The diagram shows $OCA$ and $ODB$ as radii of a circle whose centre is $O$ and whose radius is $2r$ cm. $ ngle AOB = \alpha$ radians. $CD$ and $AB$ are arcs of circles with centre $O$ and radii $r$ cm and $2r$ cm respectively. The perimeter of the shaded region $ABDC$ equals $4.4r$ cm.

Oct/Nov 2016

The diagram depicts a metal plate $ABCD$ formed from two separate parts. The part $BCD$ is a semicircle. The part $DAB$ is a segment of a circle with centre $O$ and radius $10\,\text{cm}$. Angle $BOD$ is $1.2$ radians.

Oct/Nov 2016

The diagram depicts the major arc $AB$ of a circle centred at $O$ with radius $6\text{ cm}$. The points $C$ and $D$ lie on $OA$ and $OB$ respectively, and the line $AB$ is tangent at $E$ to the arc $CED$ of a smaller circle that is also centred at $O$. Angle $COD = 1.8$ radians.

Oct/Nov 2016

The diagram displays an isosceles triangle $ABC$, with $AC = 16\text{ cm}$ and $AB = BC = 10\text{ cm}$. The circular arcs $BE$ and $BD$ are centred at $A$ and $C$ respectively, and $D$ and $E$ lie on $AC$.

Oct/Nov 2017

The diagram depicts a semicircle with centre $O$ and radius $6\text{ cm}$. Radius $OC$ is at right angles to the diameter $AB$. Point $D$ is on $AB$, and $DC$ is an arc from a circle centred at $B$.

Oct/Nov 2017

The diagram depicts a rectangle $ABCD$ where $AB = 5$ units and $BC = 3$ units. Point $P$ is on $DC$, and $AP$ is an arc of a circle with centre $B$. Point $Q$ is on $DC$, and $AQ$ is an arc of a circle with centre $D$.

Oct/Nov 2017

The diagram depicts an isosceles triangle $ACB$ with $AB = BC = 8\text{ cm}$ and $AC = 12\text{ cm}$. Arc $XC$ is taken from a circle centred at $A$ with radius $12\text{ cm}$, while arc $YC$ comes from a circle centred at $B$ with radius $8\text{ cm}$. Points $A$, $B$, $X$ and $Y$ all lie on one straight line.

Oct/Nov 2018

The diagram depicts arc $BC$ of a circle centred at $A$ with radius $5\,\text{cm}$. The arc length $BC$ is $4\,\text{cm}$. Point $D$ is positioned so that $BD$ is perpendicular to $BA$ and $DC$ is parallel to $BA$.

Oct/Nov 2018

The figure shows sector $OAC$ in a circle centred at $O$. Tangents $AB$ and $CB$ to the circle intersect at $B$. Arc $AC$ measures $6\text{ cm}$, and angle $AOC=\frac{3}{8}\pi$ radians.

Oct/Nov 2019

The diagram depicts a circle with centre $O$ and radius $r$ cm. The points $A$ and $B$ are on the circle, and angle $AOB = 2\theta$ radians. The tangents drawn to the circle at $A$ and $B$ intersect at $T$.

Oct/Nov 2019

In the diagram, $ACB$ is a semicircle with centre $O$ and radius $r$, and arc $OC$ belongs to a circle centred at $A$.

Oct/Nov 2019

The diagram displays a sector $CAB$ from a circle centred at $C$. Inside the sector is a circle with centre $O$ and radius $r$; it touches the sector at $D$, $E$ and $F$, with $COD$ forming a straight line and angle $ACD$ equal to $\theta$ radians.

Oct/Nov 2020

From the diagram, $ABC$ is an isosceles triangle with $AB = BC = r$ cm, and angle $BAC = \theta$ radians. Point $D$ is located on $AC$, and $ABD$ is a sector of a circle with centre $A$.

Oct/Nov 2020

The diagram shows arc $AB$ as part of a circle centred at $O$ with radius $8\text{ cm}$. Arc $BC$ belongs to a circle centred at A with radius $12\text{ cm}$, and $AOC$ lies on a straight line.

Oct/Nov 2020

The diagram depicts a metal plate $ABC$ whose boundary is formed by the straight side $AB$ together with the arcs $AC$ and $BC$. The length of $AB$ is $6\text{ cm}$. Arc $AC$ is an arc of a circle centred at $B$ with radius $6\text{ cm}$, and arc $BC$ is an arc of a circle centred at $A$ with radius $6\text{ cm}$.

Oct/Nov 2021

In the diagram, $AB$ and $AC$ each have length $15\text{ cm}$. Point $P$ is the point where the perpendicular from $C$ meets $AB$. Also, $CP = 9\text{ cm}$. A circle arc centred at $B$ passes through $C$ and cuts $AB$ at $Q$.

Oct/Nov 2021

The diagram displays a sector $OAB$ from a circle centred at $O$. The arc $AB$ measures 8 cm. The perimeter of the sector is stated to be 20 cm.

Oct/Nov 2022

The diagram depicts two congruent circles that overlap at points $A$ and $B$, with centres at $P$ and $Q$. Each circle has radius $r$, and the separation $PQ$ is $\frac{5}{3}r$.

Oct/Nov 2022

The diagram depicts a motif made up of the major arc $AB$ of a circle with radius $r$ and centre $O$, together with the minor arc $AOB$ of another circle that also has radius $r$ but centre $C$. Point $C$ lies on the circle centred at $O$.

Oct/Nov 2023

The diagram illustrates a coin’s outline. The arcs $AB$, $BC$ and $CA$ are each sections of circles whose centres are $C$, $A$ and $B$ respectively. $ABC$ is an equilateral triangle with side length $2\,\text{cm}$.

Oct/Nov 2023

In the diagram, points $A$, $B$ and $C$ are on a circle with centre $O$ and radius $r$. The angle $AOB$ measures $2.8$ radians. The shaded region has two arc boundaries. Its upper boundary is an arc from the circle centred at $O$ with radius $r$. Its lower boundary is an arc from a circle centred at $C$ with radius $R$.

Oct/Nov 2023

The diagram illustrates a sector of a circle, with centre $O$, for which $OB = OC = 15\text{ cm}$. The angle $BOC$ measures $\frac{2}{5}\pi$ radians. On the lines $OB$ and $OC$, points $A$ and $D$ are joined by an arc $AD$ belonging to a circle centred at $O$. The shaded part is enclosed by arcs $AD$ and $BC$ together with the straight segments $AB$ and $DC$. The area of the shaded region is given as $\frac{19}{5}\pi\text{ cm}^2$.

Oct/Nov 2024

The diagram depicts a metal plate $OABCDE$ made up of sectors from two circles, both with centre $O$. The radii of sectors $AOB$ and $EOF$ are $r$ cm, while sector $COD$ has radius $2r$ cm. The angle $AOB = \text{angle } EOF = \theta$ radians and the angle $COD = 2\theta$ radians. It is given that the plate has perimeter $14$ cm and area $10\text{ cm}^2$.

Oct/Nov 2024

The diagram depicts a metal plate $ABCDEF$ formed from five regions. Regions $BCD$ and $DEF$ are semicircles. Region $BAFO$ is a sector of a circle with centre $O$ and radius $20\text{ cm}$. $D$ is located on this circle. Regions $OBD$ and $ODF$ are triangles. The angles $BOD$ and $DOF$ are each $\theta$ radians.

Oct/Nov 2024

The diagram contains a circle centred at A with radius $r$ that passes through B, C and D. There is also a larger circle, with centre C and radius $s$, which passes through B and D. Also, the length $BD$ is $s$.

Oct/Nov 2025

The diagram illustrates a section of a circle with centre $O$ and radius $4\text{ cm}$. The chord $PQ$ measures $4\sqrt{3}\text{ cm}$, and $POQ=\theta$ radians. Point $X$ is located on the circle.

Oct/Nov 2025

The diagram illustrates a company’s new logo design. The circle sector, with centre $O$, has radius $r$ cm. The acute angle $AOC$ is $\frac{1}{3}\pi$ radians. The quadrilateral $OABC$ is a rhombus.

Oct/Nov 2025