Circles - Mensur

The questions on this page focus on:

2. Area of a sector.

3. Area of a segment.

The relevant formulae to use :	when the angle is given in degrees are	when the angle is given in radians:
Arc length		L = rθ
Area of a sector
Area of a segment	A = area of sector - area of triangle.

	where: θ is the angle in degrees and r is the radius.	where: θ is the angle in radians.

Arc length, radius and angles.	1. The angle subtended at the centre O of a sector is 42°. The radius of the circle is 10 cm. Find the arc length to the nearest mm. Answer.7.3 cm.
	2. What is the length of the arc subtended by an angle of 120° in a circle of radius 26 m? Answer.54.5 m.
	3. What is the length of the arc subtended by an angle of in a circle of radius 12 cm?
	4. The hour hand of a clock is 10 cm long. Find the length of the arc (nearest mm) through which the hour hand will turn in 5 hours. Answer.26.2 cm.
	5. A train travels around a circular arc of radius 250 metres. It travels at a speed of 15 km/hour and it takes 30 seconds to complete this curve. (i) Find the distance the train travels around this curve. (ii) Find, in radians, the angle through which the train travels. Answer.(i) At 250 m/min, distance = 125 m. (ii) θ = 0.5 rads.
	6. Compute the length (to the nearest metre) of the arc formed on Earth by 1 minute of change in the latitude. Assume the radius of Earth is 6,400 km. Answer.1862 m (nearest metre).
Find angle.	7. A circle has radius of 5 cm. Find the size of the angle subtended at the centre of this circle by an arc of length 10 cm. Answer correct to the nearest minute. Answer.Angle = 114° 35'.
	8. An arc of length 3.163 m is located on the circumference of a circle which has a radius of 12.6 m. What angle (to the nearest minute) is subtended by this arc at the centre? Answer.Angle = 14° 23'.
Find radius.	9. What is the radius of a circle (to the nearest cm) whose center angle of 98° subtends an arc on the circumference of 42 cm. Answer.Radius = 25 cm.
	10. An angle of 60° 22' subtends an arc of length 46 cm on the circumference of a circle. What is the radius of the circle (nearest mm)? Answer.Radius = 39.7 cm.
	11. An angle of 2.3 radians subtends an arc of length 35 cm on the circumference of a circle. Find the radius of the circle (nearest mm). Answer.Radius = 15.2 cm.
Miscellaneous.	12. A straight road was constructed to cut out a dangerous bend on a country road. Engineers had previously designed the bend to be part of an arc of radius 170 m. The new straight road is to have a length of 250 m. (i) Use the cosine rule to find the size of the angle θ (correct to the nearest degree). (ii) Find the distance by which the old road was shortened. (correct to the nearest metre). Answer.(i) 95°. (ii) Shortened by 32 m (282 - 250 m).
	13. A wheel of radius 40 cm and centre O rolls along a horizontal path as shown below. P is the point of contact between the wheel and the path before the wheel starts to roll. (i) Through what angle (in degrees) has the wheel rolled about O when the wheel has rolled 1 metre from its initial position? (ii) What would be the height of point P above the path when the wheel has rolled 1 metre from its initial position (to the nearest cm)? Hint:Always be careful with units. Answer.(i) Angle = 143°14'. (ii) Height = 40 + 32 = 72 cm.
Area of a sector	14. The area of a sector is 10π units² while the angle at the centre of the circle is 45°. Find the exact value of the radius of the circle. AnswerRadius = 4√5 cm.
	15. AOB is the sector of a circle with centre O and radius r. < AOB = θ. The area of the sector is 20π cm² and the length of the arc AB is 2π cm. (i) Find the radius of the circle. (ii) Calculate the size of <AOB in degrees. Answer(i) Radius = 20 cm. (ii) Angle = 17° 46'.
	16. A 10 cm arc on a circle subtends an angle of π/4 at the centre. (i) Find the exact value of the radius of the circle. (ii) Find the exact area of the major sector. Answer(i) Radius = 40/π (ii) Area = 1400/π.
	17. In the diagram, △ ABC is an equilateral triangle with sides of length 6 cm. An arc with centre A and BC as tangent, cuts AB and AC at X and Y respectively. (i) Show that the radius of the arc is cm. (ii) Find the area of the shaded portion in exact form. Answer.(ii) Area = 9(√3 - π/2) cm².
Mixed exercises	18. The circle shown in the diagram has a center at O and a radius of 70 cm. The length of the arc AB is 30 cm. (i) Show that <AOB = 0.4286 rads . (ii) Show that the shaded area represents about 3% of the area of the sector OAB.
	19. The length of an arc is 8 cm and the area of the sector is 25 cm² when an angle of θ is subtended at the center of the circle. (i) Find the radius of the circle and the angle θ at the centre (correct to 2 decimal places). (ii) Find the length of the chord cutting off the arc (correct to the nearest millimetre). Hint.Great question for learning technique. Write out the two equations for Arc Length and for Area of the sector. We have simultaneous equations with the angle and the radius as unknowns. DIVIDE the area equation by the arc length equation to isolate radius. Then substitute to find the angle. REMEMBER THIS approach for a number of topics especially series!!! Answer.(i) radius = 6.25 cm; θ = 1.28 rads. (ii) Chord is 7.5 cm.
	20. A circle with center at O has a radius of 6 cm. The area of a sector AOB in the circle has an area of 27 cm². (i) Find the angle AOB to the nearest degree and in radians (to 1 decimal place). (ii) Find the length of the arc AB. Answer.(i) Angle AOB = 86° or 1.5 rads. (ii) Arc length AB = 9 cm.
	21. From a circle with centre O, a sector OXY is cut out. The sector has radii of OX = OY = r cm. The angle subtended at the centre <XOY = θ radians where θ = 2π/3. The length of the arc XY is 10π cm. (i) Draw a diagram showing this information. (ii) Find the exact length of the chord XY. Answer.(ii) As r = 15 cm, XY = 15√3 cm.
	22. A set of three concentric circles with centre O and radii 3.5 cm, 9 cm and 12.5 cm is shown in the diagram below. The length of arc ACB is 22.5 cm (i) Calculate the size of the reflex angle <AOB (express your answer in both degrees and radians). (ii) Calculate the area of the shaded region (to nearest cm²). Answer.(i) Angle AOB = 4.483 rads. (ii) Area = 154 cm².