Circle theorems – Year 11 GCSE maths worksheet with answers to download as PDF

In each part, \(A\), \(B\) and \(C\) are points on a circle with centre \(O\).

1. Angle \(ACB=38^\circ\). Find angle \(AOB\). Give a reason. [2 marks]
2. Angle \(AOB=130^\circ\). Find angle \(ACB\). Give a reason. [2 marks]
3. Angle \(ACB=x\) and angle \(AOB=(3x-20)^\circ\). Form an equation and find \(x\). [3 marks]

\(AB\) is a diameter of a circle and \(C\) is a point on the circumference. Angle \(BAC=34^\circ\).

1. Write down angle \(ACB\) and give a reason. [2 marks]
2. Work out angle \(ABC\). [2 marks]
3. \(M\) is the centre of the circle. Show that triangle \(AMC\) is isosceles and find angle \(AMC\). [3 marks]

\(A\), \(B\), \(C\) and \(D\) are on a circle. \(AB\) and \(CD\) are chords that cross. Angle \(ABD=42^\circ\), angle \(CAD=35^\circ\) and angle \(BAC=28^\circ\).

1. Find angle \(ACD\) and give a reason. [2 marks]
2. Find angle \(CBD\). [2 marks]
3. Find angle \(BDC\). [2 marks]

\(ABCD\) is a cyclic quadrilateral.

1. Angle \(A=108^\circ\). Find angle \(C\). [2 marks]
2. Angle \(B=85^\circ\). Find angle \(D\). [1 mark]
3. Angle \(B=(2x+10)^\circ\) and angle \(D=(3x-20)^\circ\). Find \(x\) and both angles. [4 marks]

\(O\) is the centre of a circle. \(TA\) and \(TB\) are tangents to the circle at \(A\) and \(B\).

1. Angle \(AOT=62^\circ\). Find angle \(ATO\). [3 marks]
2. Angle \(ATB=50^\circ\). Find angle \(TAB\). [2 marks]
3. Using the same figure, find angle \(OAB\) and angle \(AOB\). [3 marks]

\(PQ\) is a tangent to a circle at \(A\). \(B\) and \(C\) are points on the circle. Angle \(PAB=62^\circ\) and angle \(QAC=48^\circ\).

1. Find angle \(ACB\) and give a reason. [2 marks]
2. Find angle \(ABC\). [2 marks]
3. Find angle \(BAC\). [2 marks]

Three posts \(A\), \(B\) and \(C\) stand on the edge of a circular lake. A buoy \(O\) marks the centre. The angle at the buoy between the lines to \(A\) and \(B\) is \(112^\circ\).

1. A walker stands at \(C\). What angle does she see between the lines of sight to \(A\) and \(B\)? [2 marks]
2. A second walker stands at \(D\) on the shore, on the other side of \(AB\) from \(C\). Find the angle \(ADB\). [3 marks]
3. Explain why every walker standing on the arc \(ACB\) sees the same angle. [2 marks]

A circular pizza has diameter 30 cm. A straight cut \(AC\) goes from one point \(A\) on the edge to another point \(C\) on the edge, and the diameter is \(AB\). The cut \(AC\) is 18 cm long.

1. Explain why angle \(ACB\) is a right angle. [1 mark]
2. Work out the length \(BC\). [3 marks]
3. Work out the area of triangle \(ABC\). [2 marks]
4. The pizza is cut into 8 equal slices. What is the angle at the centre of each slice? [1 mark]

A circular pond has centre \(O\) and radius 6 m. A lamp post at \(T\) is 10 m from \(O\). A string is pulled tight from \(T\) to touch the pond at \(A\) so that \(TA\) is a tangent.

1. Explain why triangle \(OAT\) has a right angle at \(A\). [1 mark]
2. Work out the length \(TA\). [3 marks]
3. A second string touches the pond at \(B\) on the other side. What is the length \(TB\) and why? [2 marks]
4. Find angle \(ATO\) to 1 decimal place. [2 marks]

A circular roundabout has radius 10 m. A straight path is cut across it as a chord \(AB\) of length 16 m. \(M\) is the midpoint of \(AB\) and \(O\) is the centre of the roundabout.

1. Explain why \(OM\) is perpendicular to \(AB\). [2 marks]
2. Work out the distance \(OM\). [3 marks]
3. Work out the area of triangle \(OAB\). [2 marks]
4. Work out angle \(AOB\) to 1 decimal place. [3 marks]

A bridge has a circular arc. The chord at water level is 24 m wide and the highest point of the arch is 4 m above the water. The centre \(O\) of the circle is below the water.

1. Let the radius be \(r\). Write down the distance from \(O\) to the chord. [1 mark]
2. Use Pythagoras to show that \(r^2=12^2+(r-4)^2\). [2 marks]
3. Solve the equation to find \(r\). [3 marks]
4. How far below the water is the centre \(O\)? [1 mark]

A circular stage has four spotlights at \(A\), \(B\), \(C\) and \(D\) on its edge, in that order. Angle \(DAB=(4x+5)^\circ\) and angle \(BCD=(2x+25)^\circ\).

1. Explain why \(4x+5+2x+25=180\). [1 mark]
2. Solve the equation. [2 marks]
3. Find both angles. [2 marks]
4. Angle \(ABC=100^\circ\). Find angle \(ADC\) and explain how you know. [2 marks]

A lighthouse is at the centre \(O\) of a circular danger zone of radius 5 km. A ship at \(S\) is 13 km from the lighthouse and sails along a straight line that just touches the edge of the zone at \(P\).

1. Explain why angle \(OPS=90^\circ\). [1 mark]
2. Work out how far the ship has sailed from \(S\) to \(P\). [3 marks]
3. Find angle \(OSP\) to 1 decimal place. [2 marks]
4. The ship sails at 24 km/h. How long does it take to reach \(P\) in minutes? [2 marks]

A football goal has posts at \(A\) and \(B\). A coach marks a circle through \(A\) and \(B\) on the pitch. A striker at \(P\) on the circle sees the goal under an angle \(APB=40^\circ\). The circle has centre \(O\).

1. Work out angle \(AOB\) and give a reason. [2 marks]
2. A second striker stands at \(Q\) on the same arc as \(P\). What angle does she see? [2 marks]
3. A third striker stands at \(R\) on the other side of \(AB\) on the circle. Work out angle \(ARB\). [2 marks]
4. The goal is 7.32 m wide. Work out the radius of the circle to 2 decimal places. [3 marks]

\(O\) is the centre of a circle. Triangle \(OAB\) has \(OA=OB\) (radii) and angle \(AOB=100^\circ\). \(TA\) and \(TB\) are tangents to the circle at \(A\) and \(B\).

1. Find angle \(OAB\) and explain how you know. [2 marks]
2. Find angle \(TAB\). [2 marks]
3. Find angle \(ATB\). [2 marks]
4. Show that \(TA=TB\) and describe quadrilateral \(OATB\). [3 marks]

\(A\), \(B\) and \(C\) lie on a circle with centre \(O\). \(TA\) is a tangent at \(A\). Angle \(TAB=64^\circ\).

1. Find angle \(ACB\) and give a reason. [2 marks]
2. Find angle \(AOB\) and give a reason. [2 marks]
3. Find angle \(OAB\) and show that it agrees with the tangent-radius angle. [4 marks]

\(A\), \(B\), \(C\) and \(D\) are on a circle with centre \(O\). Angle \(ABC=x\).

1. Write the reflex angle \(AOC\) in terms of \(x\) and give a reason. [2 marks]
2. Write the angle \(AOC\) (the other angle at \(O\)) in terms of \(x\). [2 marks]
3. Write angle \(ADC\) in terms of \(x\) and so prove that opposite angles of a cyclic quadrilateral add to \(180^\circ\). [4 marks]

\(AB\) is a diameter of a circle with centre \(O\). \(C\) and \(D\) are points on the circle on opposite sides of \(AB\). Angle \(CAB=28^\circ\) and angle \(ABD=35^\circ\).

1. Find angle \(ACB\). [1 mark]
2. Find angle \(ABC\). [1 mark]
3. Find angle \(BAD\), giving a reason. [3 marks]
4. Find angle \(ADC\), giving reasons. [4 marks]

Two circles have the same centre \(O\). The radius of the smaller circle is 5 cm and the radius of the larger circle is 13 cm. A chord \(AB\) of the larger circle touches the smaller circle at \(M\).

1. Explain why \(M\) is the midpoint of \(AB\). [2 marks]
2. Work out the length \(AB\). [3 marks]
3. Work out the area of the ring between the two circles in terms of \(\pi\). [2 marks]

\(AB\) is the diameter of a circle and \(C\) is on the circle with \(AC=10\) cm and \(BC=24\) cm.

1. Show that the radius of the circle is 13 cm. [3 marks]
2. Find the shortest distance from \(C\) to the diameter \(AB\). [4 marks]
3. Explain why no point on the circle gives a triangle with base \(AB\) and area greater than \(13\times13\) cm². [3 marks]