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Amy Ferguson Moncure

IB AA · Unit 2 · 18A Radian Measure

Find the Radius or the Angle

Start from l = rθ or from the sector area. Rearrange before you substitute.

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Key Ideas Easy Medium Hard

l = rθ

A = 12 r² θ

P = 2r + rθ

What to Remember

Rearrange

l = rθ

r = lθ

θ = lr

A = 12 r² θ, so r² = 2Aθ and θ = 2Ar²

Perimeter of a sector

The outline is two radii and the arc.

P = 2r + rθ = r(2 + θ)

r = P2 + θ

Easy

One missing value

Write the formula, rearrange, then substitute.

Example
An arc of length 12 cm is on a circle of radius 8 cm. Find the angle in radians and the area of the sector.

l = rθ

θ = lr

θ = 128

θ = 1.5

A = 12 r² θ

A = 12 × 8² × 1.5

A = 48 cm²

Answer: θ = 1.5, area 48 cm²

1
An arc has length 9.6 cm and subtends an angle of 1.5 radians. Find the radius.
Show Answer

l = rθ

r = lθ

r = 9.61.5

r = 6.4 cm

Answer: 6.4 cm

2
An arc has length 14 cm and subtends an angle of 2 radians. Find the radius.
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l = rθ

r = lθ

r = 142

r = 7 cm

Answer: 7 cm

3
A sector has area 48 cm² and angle 1.5 radians. Find the radius.
Show Answer

A = 12 r² θ

r² = 2Aθ

r² = 961.5

r² = 64

r = 8 cm

Answer: 8 cm

4
A sector has area 20 cm² and angle 2.5 radians. Find the radius.
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A = 12 r² θ

r² = 2Aθ

r² = 402.5

r² = 16

r = 4 cm

Answer: 4 cm

5
An arc of length 8 cm is on a circle of radius 5 cm. Find the angle in radians and the area of the sector.
Show Answer

l = rθ

θ = lr

θ = 85

θ = 1.6

A = 12 r² θ

A = 12 × 5² × 1.6

A = 20 cm²

Answer: θ = 1.6, area 20 cm²

Medium

Use the perimeter

A sector’s outline is two radii and the arc.

Example
A sector has perimeter 21 cm and angle 1.5 radians. Find the radius.

P = 2r + rθ

P = r(2 + θ)

r = P2 + θ

r = 213.5

r = 6 cm

Answer: 6 cm

1
A sector has perimeter 20 cm and angle 2 radians. Find the radius.
Show Answer

P = 2r + rθ

P = r(2 + θ)

r = P2 + θ

r = 204

r = 5 cm

Answer: 5 cm

2
A sector has perimeter 14 cm and angle 0.8 radians. Find the radius.
Show Answer

P = 2r + rθ

P = r(2 + θ)

r = P2 + θ

r = 142.8

r = 5 cm

Answer: 5 cm

3
A sector has perimeter 16 cm and angle 1.2 radians. Find the radius.
Show Answer

P = 2r + rθ

P = r(2 + θ)

r = P2 + θ

r = 163.2

r = 5 cm

Answer: 5 cm

4
A sector has area 40 cm² and angle 0.8 radians. Find the radius and the perimeter.Sector radius 10 cm, angle 0.810 cm0.8
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A = 12 r² θ

r² = 2Aθ

r² = 800.8

r = 10 cm

P = 2r + rθ

P = 2 × 10 + 10 × 0.8

P = 28 cm

Answer: r = 10 cm, perimeter 28 cm

5
A sector has area 45 cm² and angle 0.9 radians. Find the radius and the perimeter.Sector radius 10 cm, angle 0.910 cm0.9
Show Answer

A = 12 r² θ

r² = 2Aθ

r² = 900.9

r = 10 cm

P = 2r + rθ

P = 2 × 10 + 10 × 0.9

P = 29 cm

Answer: r = 10 cm, perimeter 29 cm

Hard

Two possibilities

An area and a perimeter can give two radii. Check that the angle is one turn or less.

Example
A sector has area 20 cm² and perimeter 18 cm. Find the possible radius.

A = 12 r² θ

P = r(2 + θ)

θ = Pr − 2

A = 12 r² (Pr − 2)

2A = P r − 2 r²

2 r² − 18 r + 40 = 0

r² − 9 r + 20 = 0

(r − 4)(r − 5) = 0

r = 4 or r = 5

When r = 4, θ = 2.5, which is less than one full turn.

When r = 5, θ = 1.6, which is less than one full turn.

Answer: 4 cm or 5 cm

1
A sector has area 15 cm² and perimeter 16 cm. Find the possible radius.
Show Answer

A = 12 r² θ

P = r(2 + θ)

θ = Pr − 2

A = 12 r² (Pr − 2)

2A = P r − 2 r²

2 r² − 16 r + 30 = 0

r² − 8 r + 15 = 0

(r − 3)(r − 5) = 0

r = 3 or r = 5

When r = 3, θ = 3.33, which is less than one full turn.

When r = 5, θ = 1.2, which is less than one full turn.

Answer: 3 cm or 5 cm

2
A sector has area 24 cm² and perimeter 20 cm. Find the possible radius.
Show Answer

A = 12 r² θ

P = r(2 + θ)

θ = Pr − 2

A = 12 r² (Pr − 2)

2A = P r − 2 r²

2 r² − 20 r + 48 = 0

r² − 10 r + 24 = 0

(r − 4)(r − 6) = 0

r = 4 or r = 6

When r = 4, θ = 3, which is less than one full turn.

When r = 6, θ = 1.33, which is less than one full turn.

Answer: 4 cm or 6 cm

3
A sector has area 12 cm² and perimeter 14 cm. Find the possible radius.
Show Answer

A = 12 r² θ

P = r(2 + θ)

θ = Pr − 2

A = 12 r² (Pr − 2)

2A = P r − 2 r²

2 r² − 14 r + 24 = 0

r² − 7 r + 12 = 0

(r − 3)(r − 4) = 0

r = 3 or r = 4

When r = 3, θ = 2.67, which is less than one full turn.

When r = 4, θ = 1.5, which is less than one full turn.

Answer: 3 cm or 4 cm

4
A sector has area 10 cm² and perimeter 20 cm. Find the possible radius.
Show Answer

A = 12 r² θ

P = r(2 + θ)

θ = Pr − 2

A = 12 r² (Pr − 2)

2A = P r − 2 r²

2 r² − 20 r + 20 = 0

r² − 10 r + 10 = 0

r = 8.87 or r = 1.13

When r = 1.13, θ = 15.7, which is more than one full turn, so it is not a single sector.

The radius is 8.87 cm, and θ = 0.254.

Answer: 8.87 cm

5
Two sectors share the angle 1.5 radians. The smaller radius is 8 cm. The larger radius is 4 cm longer. The shaded area between them is 60 cm². Find that extra length.Two sectors with the same angle8 cm4 cm wider1.5
Show Answer

Shaded area = 12 θ (R² − r²)

60 = 12 × 1.5 × ((8 + x)² − 8²)

(8 + x)² − 64 = 80

(8 + x)² = 144

8 + x = 12

x = 4 cm

Answer: 4 cm

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