A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 5: Radians · Lesson 1

Radian measure and arc length

112 marks

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Date

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Quick questions to start.

A1
Needed today
Write $13 5^{\circ}$ in radians as a multiple of $π$ .
[2 marks]
Answer
A2
Common slip
A student works out $cos 2$ on a calculator in degree mode and gets $0.9994$ . Find $cos 2$ , where the angle is $2$ radians, to 3 decimal places.
[2 marks]
Answer
A3
Needed today
An arc of a circle of radius $9$ cm subtends an angle of $4 0^{\circ}$ at the centre. Find the exact length of the arc, using a fraction of the circumference.
[2 marks]
Answer
A4
Needed today
Find the exact value of $tan 6 0^{\circ} \times cos 3 0^{\circ}$ .
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

(a) Write $21 0^{\circ}$ in radians, as a multiple of $π$ . (b) Write $1.2$ radians in degrees, to 1 decimal place.

Hint

18 0^{\circ} = π

radians. Multiply by

\frac{π}{180}

to go to radians, and by

\frac{180}{π}

to go to degrees.

B1
(a) Write $33 0^{\circ}$ in radians, as a multiple of $π$ . (b) Write $2.5$ radians in degrees, to 1 decimal place.
[3 marks]
Answer

Example 2

A sector of a circle has radius $6$ cm and arc length $7.5$ cm. Find the angle of the sector, in radians, and the perimeter of the sector.

Hint Use

s = r θ

with

θ

in radians. The perimeter is the arc and two radii.

B2
A sector has angle $0.8$ radians and perimeter $21$ cm. Find the radius and the arc length.
[4 marks]
Answer

Example 3

The chord $A B$ of a circle with centre $O$ and radius $10$ cm subtends an angle of $2$ radians at $O$ . Find the perimeter of the minor segment cut off by $A B$ , to 3 significant figures.

Hint The segment is bounded by the arc and the chord. Split triangle

O A B

into two right-angled triangles to find the chord.

B3
The chord $P Q$ of a circle with centre $O$ and radius $8$ cm subtends an angle of $1.4$ radians at $O$ . Find the perimeter of the minor segment cut off by $P Q$ , to 3 significant figures.
[5 marks]
Answer

Example 4

A circle with centre $O$ has radius $5$ cm. The chord $A B$ has length $8$ cm. Find the angle $A O B$ in radians, to 3 decimal places, and the length of the minor arc $A B$ .

Hint Use the cosine rule in triangle

O A B

, with the calculator in radian mode.

B4
A circle with centre $O$ has radius $13$ cm. The chord $A B$ has length $10$ cm. Find (a) the angle $A O B$ in radians, to 3 decimal places, (b) the length of the minor arc $A B$ , (c) the perimeter of the minor segment.
[6 marks]
Answer

Example 5

The minute hand of a clock is $12$ cm long. Find the exact distance moved by its tip in $40$ minutes.

Hint One hour is a full turn,

2 π

radians. Find the angle first.

B5
The minute hand of a clock is $15$ cm long. Find the exact distance moved by its tip in $25$ minutes.
[3 marks]
Answer

Practice

A-Level

C1
Ola finds the arc length of a sector of radius $6$ cm and angle $4 0^{\circ}$ . She writes: "arc $= r θ = 6 \times 40 = 240$ cm". Find her mistake and give the correct arc length exactly.
[4 marks]
Answer

C2
The string of a pendulum is $80$ cm long. The pendulum swings $0.3$ radians either side of the vertical. Find (a) the length of the arc the bob travels in one swing from one end to the other, (b) the height the bob rises above its lowest point, to 3 significant figures.
[5 marks]
Answer
C3
A shape is bounded by two arcs with the same centre, of radii $8$ cm and $5$ cm, both with angle $θ$ radians, and two straight edges of length $3$ cm. The perimeter of the shape is $30$ cm. Find the exact value of $θ$ .
[5 marks]
Answer
C4
Ben says that the perimeter of a sector of radius $9$ cm and angle $\frac{2 π}{3}$ is $6 π$ cm. Find his mistake and give the correct perimeter, exactly and to 3 significant figures.
[4 marks]
Answer
C5
Two sectors each have perimeter $30$ cm. Sector $P$ has radius $6$ cm. Sector $Q$ has angle $1.5$ radians. Find the angle of $P$ , the radius of $Q$ , and the exact difference between their arc lengths.
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
A sector $O A B$ of a circle has radius $r$ cm and angle $θ$ radians. Its perimeter is $40$ cm.
(a)
Show that $θ = \frac{40 - 2 r}{r}$ .
[2 marks]
(b)
The arc $A B$ has length $16$ cm. Find $r$ and $θ$ .
[3 marks]
Answer
(c)
Find the set of possible values of $r$ , giving the ends of the interval exactly.
[4 marks]
Answer
Total for D1: 9 marks

D2
A circle with centre $O$ has radius $6$ cm. The chord $A B$ has length $63$ cm. Zara says: "Angle $A O B$ is $12 0^{\circ}$ , so the minor arc $A B$ is $r θ = 6 \times 120 = 720$ cm."
Zara is wrong.
(a)
Explain her mistake.
[2 marks]
(b)
Show that angle $A O B = \frac{2 π}{3}$ and find the exact length of the minor arc $A B$ .
[3 marks]
Answer
(c)
Find the exact perimeter of the major segment cut off by $A B$ .
[3 marks]
Answer
Total for D2: 8 marks
D3
Two circles each have radius $5$ cm. Their centres $A$ and $B$ are $6$ cm apart. Find the perimeter of the region that is inside both circles, to 3 significant figures.
[7 marks]
Answer
D4
A belt passes tightly round two pulleys of radii $9$ cm and $3$ cm. The centres $A$ and $B$ of the pulleys are $12$ cm apart. The straight parts of the belt touch both pulleys.
(a)
Show that each straight part of the belt has length $63$ cm.
[2 marks]
(b)
Show that the belt is in contact with the larger pulley for an angle of $\frac{4 π}{3}$ radians.
[3 marks]
(c)
Find the exact length of the belt.
[5 marks]
Answer
Total for D4: 10 marks
D5
Shape 1 is made from an equilateral triangle $A B C$ of side $10$ cm. Each pair of vertices is joined by an arc whose centre is the third vertex, as in the diagram. Shape 2 is made in the same way from a regular pentagon of side $6$ cm: each pair of next-door vertices is joined by an arc whose centre is the vertex opposite them.
(a)
Find the exact perimeter of Shape 1.
[3 marks]
Answer
(b)
In the pentagon of Shape 2, each vertex is joined to the two vertices opposite it. Show that the angle between these two lines is $\frac{π}{5}$ .
[3 marks]
(c)
Find the perimeter of Shape 2, to 3 significant figures.
[3 marks]
Answer
Total for D5: 9 marks
D6
A wheel of radius $30$ cm rolls along level ground without slipping. A point $P$ on the rim is touching the ground at the start.
(a)
The wheel rolls $1$ m. Find the angle, in radians, through which it turns.
[2 marks]
Answer
(b)
Find the height of $P$ above the ground at this moment, to 3 significant figures.
[3 marks]
Answer
(c)
Find the exact distance the wheel rolls before $P$ is first $45$ cm above the ground.
[3 marks]
Answer
Total for D6: 8 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Three circles each have radius $2$ cm. Their centres $A$ , $B$ and $C$ are the corners of a triangle with sides $5$ cm, $6$ cm and $7$ cm. A band is stretched tightly round the outside of all three circles. Show that the length of the band is $18 + 4 π$ cm.
[9 marks]
Hint 1 · What to try
Each straight part of the band touches two circles of the same radius. How long is it, compared with a side of the triangle?
Hint 2 · The first line
At each circle the band follows an arc. The radii at the two ends of that arc are perpendicular to the two sides that meet at that centre.
Hint 3 · The full method
The arc at a corner turns through the exterior angle of the triangle there. What do the three exterior angles of a triangle add up to?
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can change an angle between degrees and radians, exactly or to a given accuracy.
I can use s = rθ with θ in radians to find an arc length, a radius or an angle.
I can find the perimeter of a sector or a segment, including the straight edges.
I can find an angle at the centre from a chord, using the cosine rule or right-angled triangles.

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