Circular measure: past paper questions

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• Question 7Oct/Nov 2025, paper 11Structured8 marksHas a diagram

  The diagram shows a sector of a circle with centre $O$ and radius $r\text{ cm}$. The shaded region is bounded by the chord $AB$ and the arc $AB$. The size of angle $AOB$ is $\frac{2}{3}\pi \text{ radians}$.

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• Question 3Oct/Nov 2024, paper 11Structured5 marksHas a diagram

  The diagram shows a sector of a circle, centre $O$, where $OB=OC=15\text{ cm}$. The size of angle $BOC$ is $\frac{2}{5}\pi \text{ radians}$. Points $A$ and $D$ on the lines $OB$ and $OC$ respectively are joined by an arc $AD$ of a circle with centre $O$.

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• Question 6Oct/Nov 2023, paper 11Structured7 marksHas a diagram

  The diagram shows a motif formed by the major arc $AB$ of a circle with radius $r$ and centre $O$, and the minor arc $AOB$ of a circle, also with radius $r$ but with centre $C$. The point $C$ lies on the circle with centre $O$.

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• Question 7May/June 2026, paper 13Structured6 marks

  The diagram shows the design for a garden. The points $A$, $B$ and $C$ form the vertices of an equilateral triangle of side length $r$ metres. The arcs $AC$ and $AB$ are parts of circles with centres $B$ and $C$ respectively and are both of radius $r$ metres.

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• Question 10Oct/Nov 2025, paper 12Structured9 marksHas a diagram

  The diagram shows a circle with centre $A$ and radius $r$ passing through points $B$, $C$ and $D$. A larger circle of radius $s$ has centre $C$ and passes through $B$ and $D$. The length $BD$ is also $s$.

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• Question 6Oct/Nov 2024, paper 12Structured6 marks

  The diagram shows a metal plate $OABCDEF$ consisting of sectors of two circles, each with centre $O$. The radii of sectors $AOB$ and $EOF$ are $r\text{ cm}$ and the radius of sector $COD$ is $2r\text{ cm}$.

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• Question 4Oct/Nov 2023, paper 12Structured6 marks

  The diagram shows the shape of a coin. The three arcs $AB$, $BC$ and $CA$ are parts of circles with centres $C$, $A$ and $B$ respectively. $ABC$ is an equilateral triangle with sides of length $2\text{ cm}$.

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• Question 5Oct/Nov 2025, paper 13Structured5 marksHas a diagram

  The diagram shows part of a circle with centre $O$ and radius $4\text{ cm}$. The chord $PQ$ is of length $4\sqrt{3}\text{ cm}$ and angle $POQ=\theta \text{ radians}$. The point $X$ lies on the circle.

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• Question 7Oct/Nov 2024, paper 13Structured10 marksHas a diagram

  The diagram shows a metal plate $ABCDEF$ consisting of five parts. The parts $BCD$ and $DEF$ are semicircles. The part $BAFO$ is a sector of a circle with centre $O$ and radius $20\text{ cm}$, and $D$ lies on this circle.

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• Question 9May/June 2025, paper 11Structured8 marksHas a diagram

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

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• Question 7May/June 2024, paper 11Structured8 marksHas a diagram

  In the diagram, $AOD$ and $BC$ are two parallel straight lines. Arc $AB$ is part of a circle with centre $O$ and radius $15\text{ cm}$. Angle $BOA=\theta \text{ radians}$.

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• Question 4May/June 2023, paper 11Structured4 marksHas a diagram

  The diagram shows a sector $ABC$ of a circle with centre $A$ and radius $8\text{ cm}$. The area of the sector is $\frac{16}{3}\pi {\text{ cm}}^2$. The point $D$ lies on the arc $BC$.

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• Question 8May/June 2025, paper 12Structured9 marksHas a diagram

  The diagram shows the circle with equation $x^2+y^2-14x+8y+36=0$ and the line $y=-2$. The line intersects the circle at the points $A$ and $B$. The centre of the circle is $C$.

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• Question 8May/June 2024, paper 12Structured10 marksHas a diagram

  The diagram shows a symmetrical plate $ABCDEF$. The line $ABCD$ is straight and the length of $BC$ is $2\text{ cm}$. Each of the two sectors $ABF$ and $DCE$ is of radius $r\text{ cm}$ and each of the angles $ABF$ and $DCE$ is equal to $\frac{1}{3}\pi \text{ radians}$.

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• Question 6May/June 2023, paper 12Structured6 marksHas a diagram

  The diagram shows a sector $OAB$ of a circle with centre $O$. Angle $AOB=\theta$ radians and $OP=AP=x$. Show that the arc length $AB$ is $2x\theta \cos \theta$.

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• Question 8May/June 2025, paper 13Structured8 marks

  The diagram shows a square $ABCD$ where each side has length $12\text{ cm}$. Points $E$ and $F$ lie on the sides $BC$ and $CD$ respectively and are such that $BE=\frac{1}{3}BC$ and $DF=\frac{1}{3}DC$. The arc $EF$ is part of a circle with centre $A$.

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• Question 3May/June 2024, paper 13Structured6 marksHas a diagram

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

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• Question 6May/June 2023, paper 13Structured7 marksHas a diagram

  The diagram shows a sector $OAB$ of a circle with centre $O$ and radius $r\text{ cm}$. Angle $AOB=\theta \text{ radians}$. It is given that the length of the arc $AB$ is $9.6\text{ cm}$ and that the area of the sector $OAB$ is $76.8{\text{ cm}}^2$.

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