Chapter 8: Circular Measure
8.1 Radian Measure – Definition and Conversion
Definition:
One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.
\(1 \text{ radian} = \dfrac{180^\circ}{\pi} \approx 57.3^\circ\)
Conversion formulas:
\(\theta \text{ (radians)} = \theta \text{ (degrees)} \times \dfrac{\pi}{180}\)
\(\theta \text{ (degrees)} = \theta \text{ (radians)} \times \dfrac{180}{\pi}\)
Proof: Relationship between degrees and radians
The circumference of a circle is \(2\pi r\).
The angle subtended by the full circumference is \(360^\circ\).
So \(360^\circ\) corresponds to \(2\pi\) radians.
Therefore:
\[
360^\circ = 2\pi \text{ rad} \Rightarrow 1^\circ = \frac{2\pi}{360} = \frac{\pi}{180} \text{ rad}
\]
Thus:
\[
\theta \text{ (rad)} = \theta \text{ (deg)} \times \frac{\pi}{180}
\]
Common conversions:
| Degrees |
\(0^\circ\) |
\(30^\circ\) |
\(45^\circ\) |
\(60^\circ\) |
\(90^\circ\) |
\(180^\circ\) |
\(270^\circ\) |
\(360^\circ\) |
| Radians |
0 |
\(\frac{\pi}{6}\) |
\(\frac{\pi}{4}\) |
\(\frac{\pi}{3}\) |
\(\frac{\pi}{2}\) |
\(\pi\) |
\(\frac{3\pi}{2}\) |
\(2\pi\) |
Worked Example 8.1 (Converting degrees to radians – non-calculator)
Convert:
- \(45^\circ\) to radians
- \(120^\circ\) to radians
- \(210^\circ\) to radians
Solution:
(a) \(45^\circ = 45 \times \frac{\pi}{180} = \frac{\pi}{4}\)
(b) \(120^\circ = 120 \times \frac{\pi}{180} = \frac{2\pi}{3}\)
(c) \(210^\circ = 210 \times \frac{\pi}{180} = \frac{7\pi}{6}\)
Worked Example 8.2 (Converting radians to degrees – non-calculator)
Convert:
- \(\frac{\pi}{6}\) to degrees
- \(\frac{3\pi}{4}\) to degrees
- \(\frac{5\pi}{3}\) to degrees
Solution:
(a) \(\frac{\pi}{6} = \frac{\pi}{6} \times \frac{180}{\pi} = 30^\circ\)
(b) \(\frac{3\pi}{4} = \frac{3\pi}{4} \times \frac{180}{\pi} = 135^\circ\)
(c) \(\frac{5\pi}{3} = \frac{5\pi}{3} \times \frac{180}{\pi} = 300^\circ\)
Worked Example 8.3 (Mauritian context – Ferris wheel angle)
A Ferris wheel at a Mauritian amusement park rotates through an angle of \(120^\circ\). What is this angle in radians?
Solution:
\(120^\circ = 120 \times \frac{\pi}{180} = \frac{2\pi}{3}\) radians.
Answer: \(\frac{2\pi}{3}\) radians.
8.2 Arc Length \(s = r\theta\)
Formula:
For a circle of radius \(r\), the length \(s\) of an arc subtending an angle \(\theta\) (in radians) at the centre is:
\[
s = r\theta
\]
Proof: Arc length formula
The circumference of a full circle is \(2\pi r\), and this corresponds to \(2\pi\) radians (a full revolution).
By proportion:
\[
\frac{s}{2\pi r} = \frac{\theta}{2\pi} \Rightarrow s = r\theta
\]
Important: The angle \(\theta\) must be in radians for this formula to work.
Worked Example 8.4 (Finding arc length – non-calculator)
A circle has radius 8 cm. Find the length of an arc subtending an angle of \(\frac{\pi}{4}\) radians at the centre.
Solution:
\[
s = r\theta = 8 \times \frac{\pi}{4} = 2\pi \text{ cm}
\]
Answer: \(2\pi\) cm.
Worked Example 8.5 (Finding arc length – degrees given)
A circle has radius 10 cm. Find the arc length for an angle of \(60^\circ\).
Solution:
First convert to radians: \(60^\circ = \frac{\pi}{3}\)
\[
s = 10 \times \frac{\pi}{3} = \frac{10\pi}{3} \text{ cm}
\]
Answer: \(\frac{10\pi}{3}\) cm.
Worked Example 8.6 (Finding angle from arc length)
An arc of length 12 cm subtends an angle at the centre of a circle with radius 5 cm. Find the angle in radians and degrees.
Solution:
\[
\theta = \frac{s}{r} = \frac{12}{5} = 2.4 \text{ radians}
\]
In degrees: \(2.4 \times \frac{180}{\pi} \approx 137.5^\circ\)
Answer: \(2.4\) radians \(\approx 137.5^\circ\).
Worked Example 8.7 (Mauritian context – circular track)
A circular running track has a radius of 50 m. A runner covers an arc of length 100 m. What angle (in radians) has the runner covered?
Solution:
\[
\theta = \frac{s}{r} = \frac{100}{50} = 2 \text{ radians}
\]
Answer: 2 radians.
8.3 Sector Area \(A = \frac{1}{2}r^2\theta\)
Formula:
For a circle of radius \(r\), the area \(A\) of a sector with angle \(\theta\) (in radians) at the centre is:
\[
A = \frac{1}{2}r^2\theta
\]
Proof: Sector area formula
The area of a full circle is \(\pi r^2\), and this corresponds to \(2\pi\) radians.
By proportion:
\[
\frac{A}{\pi r^2} = \frac{\theta}{2\pi} \Rightarrow A = \frac{1}{2}r^2\theta
\]
Important: The angle \(\theta\) must be in radians for this formula to work.
Perimeter of a sector:
\[
P = s + 2r = r\theta + 2r = r(\theta + 2)
\]
Worked Example 8.8 (Finding sector area – non-calculator)
A circle has radius 6 cm. Find the area of a sector with angle \(\frac{\pi}{3}\) radians.
Solution:
\[
A = \frac{1}{2}r^2\theta = \frac{1}{2}(6)^2\left(\frac{\pi}{3}\right) = \frac{1}{2}(36)\left(\frac{\pi}{3}\right) = 6\pi \text{ cm}^2
\]
Answer: \(6\pi\) cm\(^2\).
Worked Example 8.9 (Finding sector area – degrees given)
Find the area of a sector with radius 8 cm and angle \(45^\circ\).
Solution:
\(45^\circ = \frac{\pi}{4}\)
\[
A = \frac{1}{2}(8)^2\left(\frac{\pi}{4}\right) = \frac{1}{2}(64)\left(\frac{\pi}{4}\right) = 8\pi \text{ cm}^2
\]
Answer: \(8\pi\) cm\(^2\).
Worked Example 8.10 (Finding angle from sector area)
A sector of radius 10 cm has area 25 cm². Find the angle in radians.
Solution:
\[
25 = \frac{1}{2}(10)^2\theta = \frac{1}{2}(100)\theta = 50\theta
\]
\[
\theta = \frac{25}{50} = 0.5 \text{ radians}
\]
Answer: 0.5 radians.
Worked Example 8.11 (Mauritian context – pizza slice)
A circular pizza with radius 20 cm is cut into 8 equal slices. Find the area of one slice.
Solution:
Angle of one slice: \(\theta = \frac{2\pi}{8} = \frac{\pi}{4}\)
\[
A = \frac{1}{2}(20)^2\left(\frac{\pi}{4}\right) = \frac{1}{2}(400)\left(\frac{\pi}{4}\right) = 50\pi \text{ cm}^2
\]
Answer: \(50\pi\) cm\(^2\).
8.4 Applications and Problem Solving
Many problems combine arc length and sector area, often with:
- Perimeter of a sector = arc length + 2 radii
- Area of a segment = sector area – triangle area
- Angle problems requiring conversion
Worked Example 8.12 (Perimeter of a sector – non-calculator)
A sector of radius 5 cm has angle \(\frac{2\pi}{3}\). Find its perimeter.
Solution:
Arc length: \(s = r\theta = 5 \times \frac{2\pi}{3} = \frac{10\pi}{3}\)
Perimeter: \(P = s + 2r = \frac{10\pi}{3} + 10 = 10\left(\frac{\pi}{3} + 1\right) \text{ cm}\)
Answer: \(10\left(\frac{\pi}{3} + 1\right)\) cm.
Worked Example 8.13 (Combined arc and sector problem)
A sector has perimeter 20 cm and radius 6 cm. Find the angle in radians.
Solution:
\[
P = r\theta + 2r
\]
\[
20 = 6\theta + 12
\]
\[
6\theta = 8 \Rightarrow \theta = \frac{4}{3} \text{ radians}
\]
Answer: \(\frac{4}{3}\) radians.
Worked Example 8.14 (Mauritian context – circular garden)
A circular garden in Curepipe has a path along an arc of length 15 m. If the radius of the garden is 10 m, find:
- The angle subtended by the arc in radians
- The area of the sector that the path borders
Solution:
(a) \(\theta = \frac{s}{r} = \frac{15}{10} = 1.5\) radians
(b) \(A = \frac{1}{2}r^2\theta = \frac{1}{2}(10)^2(1.5) = \frac{1}{2}(100)(1.5) = 75\) m\(^2\)
Answer: (a) 1.5 radians, (b) 75 m\(^2\).
Chapter 8 Summary
| Concept |
Formula |
Notes |
| Radian conversion |
\(\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}\) |
Angle must be in radians for arc/sector formulas |
| Arc length |
\(s = r\theta\) |
|
| Sector area |
\(A = \frac{1}{2}r^2\theta\) |
|
| Sector perimeter |
\(P = r\theta + 2r\) |
|
| Key angles |
\(30^\circ = \frac{\pi}{6}\), \(45^\circ = \frac{\pi}{4}\), \(60^\circ = \frac{\pi}{3}\), \(90^\circ = \frac{\pi}{2}\), \(180^\circ = \pi\) |
|
Exercises – Chapter 8
Easy (Drill – Non-Calculator)
- Convert to radians:
(a) \(30^\circ\)
(b) \(135^\circ\)
(c) \(225^\circ\)
- Convert to degrees:
(a) \(\frac{\pi}{6}\)
(b) \(\frac{2\pi}{3}\)
(c) \(\frac{5\pi}{4}\)
- A circle has radius 10 cm. Find:
(a) The arc length for angle \(\frac{\pi}{5}\)
(b) The sector area for angle \(\frac{\pi}{5}\)
Medium (Examination Style)
- A sector has radius 12 cm and angle \(\frac{2\pi}{3}\) radians. Find:
(a) Arc length
(b) Sector area
(c) Perimeter of the sector
- An arc of length 24 cm subtends an angle of 0.8 radians at the centre of a circle. Find:
(a) The radius of the circle
(b) The area of the sector
- Mauritian context: A wind turbine blade of length 15 m rotates through an angle of \(120^\circ\).
(a) Convert \(120^\circ\) to radians.
(b) Find the distance travelled by the tip of the blade.
(c) Find the area swept by the blade.
- A sector has perimeter 30 cm and radius 10 cm. Find the angle in radians.
Hard (Challenge for A*)
- A sector has area \(A\) and perimeter \(P\). Express the radius \(r\) in terms of \(P\) and \(A\).
- Mauritian context: A circular lawn in a park in Port Louis has a sector removed to create a flower bed. The remaining sector has radius 12 m and perimeter 40 m. Find the angle of the sector.
- The hour hand of a clock is 8 cm long.
(a) What angle (in radians) does it turn through in 3 hours?
(b) Find the distance travelled by the tip of the hour hand in 3 hours.
(c) Find the area swept by the hour hand in 3 hours.
(d) If the minute hand is 12 cm long, how far does its tip travel in 20 minutes?