IGCSEAdditional MathsCIETopic QuestionsTrigonometryCircular Measure

Circular Measure (CIE IGCSE Additional Maths)

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1a
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q9-2025-specimen-paper-2-cie-igcse-additional-maths

The diagram shows a rectangle A B C D and an arc A X B of a circle with centre at O, the midpoint of D C.
The length of B C is 8 space cm and the length of D C is 30 space cm. Angle A O B is theta radians.

Find the perimeter of the shape A D O C B X.

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1b
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Find the area of the shape A D O C B X.

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2a
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In this question all lengths are in centimetres.

q10a-june-2021-cie-igcse-additional-maths

The diagram shows a shaded shape. The arc A B is the major arc of a circle, centre O, radius 10. The line A B is of length 15, the line O C is of length 25 and the lengths of A C and B C are equal.

Show that the angle A O B is 1.70 radians correct to 2 decimal places.

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2b
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Find the perimeter of the shaded shape.

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2c
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Find the area of the shaded shape.

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3a
Sme Calculator5 marks

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

q11-0606-w20-qp-12-additional-maths

The diagram shows the rectangle A D E F, where A F space equals space D E space equals space r. The points B and C lie on A D such that A B equals C D equals r . The curve B C is an arc of the circle, centre O, radius r and has a length of 1.5 r.

Show that the perimeter of the shaded region is left parenthesis 7.5 plus 2 space sin space 0.75 right parenthesis r .

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3b
Sme Calculator4 marks

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

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1
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A circle has a radius of 6 cm. A sector of this circle has a perimeter of 2 open parentheses 6 plus 5 pi close parentheses space cm. Find the area of this sector.

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2
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q6b-0606-m20-qp-22-additional-maths

The diagram shows the sector A O B of a circle with centre O and radius 7 cm.
Angle A O B space equals pi over 4 radians. Find the perimeter of the shaded region.

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3a
Sme Calculator1 mark

q7-0606-s20-qp-13-additional-maths
The diagram shows an isosceles triangle O A B such that O A space equals space O B space equals space 12 cm and angle A O B space equals space theta space radians.
Points C and D lie on O A and O B respectively such that C D is an arc of the circle, centre O, radius 10 space cm. The area of the sector O C D space equals space 35 cm squared .

Show that theta space equals space 0.7.

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3b
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Find the perimeter of the shaded region.

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3c
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Find the area of the shaded region.

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1a
Sme Calculator1 mark

q7-0606-s20-qp-11-additional-maths

The diagram shows an isosceles triangle O A B such that O A space equals space O B and angle A O B space equals space theta radians. The points C and D lie on O A and O B respectively. C D is an arc of length 9.6 cm of the circle, centre O, radius 12 cm. The arc C D touches the line A B at the point M.

Find the value of theta.

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1b
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Find the total area of the shaded regions.

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1c
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Find the total perimeter of the shaded regions.

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2a
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q11-0606-s20-qp-22-additional-maths

The circles with centres C subscript 1 and C subscript 2 have equal radii of length r space cm. The line C subscript 1 C subscript 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 straight pi space cm, find the value of r. [4]

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2b
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Find the exact area of the shaded region.

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3a
Sme Calculator1 mark

In this question all lengths are in centimetres.

q8-0606-w20-qp-13-additional-maths

The diagram shows the figure A B C. The arc A B is part of a circle, centre O, radius r, and is of length 1.45 space r. The point O lies on the straight line C B such that C O space equals space 0.5 space r.

Find, in radians, the angle A O B.

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3b
Sme Calculator3 marks

Find the area of A B C, giving your  in the form k r squared, where k is a constant.

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3c
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Given that the perimeter of A B C is 12 space cm, find the value of r.

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4a
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q12-0606-w20-qp-21-additional-maths

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 A C and B C are tangents to the circles, centres B and A respectively. Find the angle C A B in radians.

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4b
Sme Calculator4 marks

Find the perimeter of the whole shape.

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4c
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Find the area of the whole shape.

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