40: Radians, Arc Length and Sector Area “Teach A Level Maths” Vol. 1: AS Core Modules 40: Radians, Arc Length and Sector Area © Christine Crisp

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r r r x = 1 radian Radians Radians are units for measuring angles. They can be used instead of degrees. O 1 radian is the size of the angle formed at the centre of a circle by 2 radii which join the ends of an arc equal in length to the radius. x r r r x = 1 radian = 1 rad. or 1c

Radians If the arc is 2r, the angle is 2 radians. O r 2c r 2r

r 3r r 3c Radians If the arc is 2r, the angle is 2 radians. O r 3c 3r r

r r Radians If the arc is 2r, the angle is 2 radians. O r r

r r r Radians If the arc is 2r, the angle is 2 radians. O r r r If the arc is r, the angle is radians.

r Radians If the arc is r, the angle is radians. O But, r is half the circumference of the circle so the angle is Hence,

r r r x = 1 radian Radians Hence, O x We sometimes say the angle at the centre is subtended by the arc.

1 radian SUMMARY Radians One radian is the size of the angle subtended by the arc of a circle equal to the radius 1 radian

Exercises 1. Write down the equivalent number of degrees for the following number of radians: (a) (b) (c) (d) Ans: (a) (b) (c) (d) It is very useful to memorize these conversions 2. Write down, as a fraction of , the number of radians equal to the following: (a) (b) (c) (d) (a) (b) (c) (d) Ans:

r l . . . l is the same fraction of the circumference. So, Arc Length and Sector Area Consider a sector of a circle with angle . Let the arc length be l . O r Then, whatever fraction is of the total angle at O, . . . l . . . l is the same fraction of the circumference. So, circumference ( In the diagram this is about one-third.) circumference

r A Arc Length and Sector Area Also, the sector area A is the same fraction of the area of the circle. O r circle area A

Examples 1. Find the arc length, l, and area, A, of the sector of a circle of radius 7 cm. and sector angle 2 radians. Solution: where is in radians

Examples 2. Find the arc length, l, and area, A, of the sector of a circle of radius 5 cm. and sector angle . Give exact answers in terms of . Solution: where is in radians rads. rads. So,

1 radian SUMMARY Radians An arc of a circle equal in length to the radius subtends an angle equal to 1 radian. 1 radian For a sector of angle radians of a circle of radius r, the arc length, l, is given by the sector area, A, is given by

Exercises O 4 cm A l 1. Find the arc length, l, and area, A, of the sector shown. 2. Find the arc length, l, and area, A, of the sector of a circle of radius 8 cm. and sector angle . Give exact answers in terms of .

Exercises O 4 cm A l 1. Solution:

Exercises O 8 cm A l 2. Solution: where is in radians rads. rads. So,

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

r 1 radian SUMMARY Radians One radian is the size of the angle subtended by the arc of a circle equal to the radius 1 radian r O x

r l . . . l is the same fraction of the circumference. So, Arc Length and Sector Area Let the arc length be l . O r l Consider a sector of a circle with angle . Then, whatever fraction is of the total angle at O, . . . . . . l is the same fraction of the circumference. So, ( In the diagram this is about one-third.) circumference

r A Arc Length and Sector Area Also, the sector area A is the same fraction of the area of the circle. A circle area

SUMMARY For a sector of angle radians of a circle of radius r, the arc length, l, is given by the sector area, A, is given by