CIE AS/A Level Maths-1.4 Circular measure- Study Notes- New Syllabus - 2026-2027
CIE AS/A Level Maths-1.4 Circular measure- Study Notes- New Syllabus
Ace AS/A Level Maths Exam with CIE AS/A Level Maths-1.4 Circular measure- Study Notes
Key Concepts:
- Arc length of a circle
- Sector Area of a Circle
- Relationship Between Radians and Degrees
Arc length and Sector Area of a circle
Definition of a Radian using Arc Length
A radian is a unit of angle measure, defined using the arc length of a circle. One radian is the angle at the center of a circle when the arc length is equal to the radius of the circle.
If the arc length \(s\) of a circle with radius \(r\) subtends an angle \(\theta\) at the center, then
\(\theta = \dfrac{s}{r}\) (in radians).
Sector Area of a Circle
The area of a sector formed by an angle \(\theta\) (in radians) in a circle of radius \(r\) is:
\(A = \dfrac{1}{2} r^{2} \theta\).
Important: Again, \(\theta\) must be in radians for this formula.
Relationship Between Radians and Degrees
We know a full circle is \(360^\circ\), and the circumference of a circle is \(2 \pi r\). So, for a full revolution:
\(\theta = \dfrac{s}{r} = \dfrac{2 \pi r}{r} = 2 \pi \ \text{radians}.\)
Therefore:
- \(360^\circ = 2\pi \ \text{radians}\)
- \(180^\circ = \pi \ \text{radians}\)
This gives us the conversion formulas:
\(1^\circ = \dfrac{\pi}{180} \ \text{radians}, \quad 1 \ \text{radian} = \dfrac{180}{\pi}^\circ.\)
Example :
Find the arc length of a circle of radius \(7 \ \text{cm}\) when the central angle is \(60^\circ\).
▶️ Answer/Explanation
Step 1: Convert angle to radians
\(\theta = 60^\circ \times \dfrac{\pi}{180} = \dfrac{\pi}{3}\ \text{radians}\).
Step 2: Apply arc length formula
\(s = r\theta = 7 \times \dfrac{\pi}{3} = \dfrac{7\pi}{3}\ \text{cm}\).
Final Answer: Arc length = \(\dfrac{7\pi}{3}\ \text{cm}\).
Example:
A circle of radius \(10 \ \text{cm}\) has a central angle of \(90^\circ\). Find the area of the sector.
▶️ Answer/Explanation
Step 1: Convert angle to radians
\(\theta = 90^\circ \times \dfrac{\pi}{180} = \dfrac{\pi}{2}\ \text{radians}\).
Step 2: Apply sector area formula
\(A = \dfrac{1}{2} r^2 \theta = \dfrac{1}{2}(10^2)(\dfrac{\pi}{2}) = 50\pi \ \text{cm}^2\).
Final Answer: Area of sector = \(50\pi \ \text{cm}^2\).
Example:
The arc length of a circle of radius \(14 \ \text{cm}\) is \(21 \ \text{cm}\). Find:
- The angle subtended at the center in radians.
- The angle in degrees.
▶️ Answer/Explanation
Step 1: Recall arc length formula
\(s = r\theta \ \Rightarrow \ \theta = \dfrac{s}{r}\).
Step 2: Substitute values
\(\theta = \dfrac{21}{14} = 1.5 \ \text{radians}\).
Step 3: Convert to degrees
\(\theta = 1.5 \times \dfrac{180}{\pi} \approx 85.9^\circ\).
Final Answers:
- \(\theta = 1.5 \ \text{radians}\)
- \(\theta \approx 85.9^\circ\)
Example:
A circle has radius \(12 \ \text{cm}\). A sector of the circle has a central angle of \(150^\circ\).
Find:
- The angle in radians.
- The length of the arc.
- The area of the sector.
▶️ Answer/Explanation
Step 1: Convert angle to radians
\(\theta = 150^\circ \times \dfrac{\pi}{180} = \dfrac{5\pi}{6} \ \text{radians}\).
Step 2: Find arc length
Formula: \(s = r\theta\).
\(s = 12 \times \dfrac{5\pi}{6} = 10\pi \ \text{cm}\).
Step 3: Find sector area
Formula: \(A = \dfrac{1}{2} r^2 \theta\).
\(A = \dfrac{1}{2}(12^2)\left(\dfrac{5\pi}{6}\right)\).
\(A = 72 \times \dfrac{5\pi}{6} = 60\pi \ \text{cm}^2\).
Final Answers:
- \(\theta = \dfrac{5\pi}{6} \ \text{radians}\)
- Arc length = \(10\pi \ \text{cm}\)
- Sector area = \(60\pi \ \text{cm}^2\)
Example:
A sector of a circle has radius \(8 \ \text{cm}\) and central angle \(\dfrac{\pi}{3}\) radians.
Within this sector, a triangle is formed by joining the two radii and the chord (straight line) across the arc.
Find:
- The arc length of the sector.
- The area of the sector.
- The length of the chord.
- The area of the triangle formed by the two radii and the chord.
▶️ Answer/Explanation
Step 1: Find arc length
Formula: \(s = r\theta\).
\(s = 8 \times \dfrac{\pi}{3} = \dfrac{8\pi}{3} \ \text{cm}\).
Step 2: Find sector area
Formula: \(A = \dfrac{1}{2} r^2 \theta\).
\(A = \dfrac{1}{2}(8^2)(\dfrac{\pi}{3}) = 32 \times \dfrac{\pi}{3} = \dfrac{32\pi}{3} \ \text{cm}^2\).
Step 3: Find chord length
The triangle formed is isosceles with two sides of length \(r = 8\), and included angle \(\theta = \dfrac{\pi}{3}\).
Using cosine rule: \(c^2 = 8^2 + 8^2 – 2(8)(8)\cos(\dfrac{\pi}{3})\).
\(c^2 = 64 + 64 – 128 \times \dfrac{1}{2} = 128 – 64 = 64\).
\(c = 8 \ \text{cm}\).
Step 4: Find triangle area
Formula: \(\text{Area} = \dfrac{1}{2}ab \sin C\).
\(\text{Area} = \dfrac{1}{2}(8)(8)\sin(\dfrac{\pi}{3})\).
= \(32 \times \dfrac{\sqrt{3}}{2} = 16\sqrt{3} \ \text{cm}^2\).
Final Answers:
- Arc length = \(\dfrac{8\pi}{3} \ \text{cm}\)
- Sector area = \(\dfrac{32\pi}{3} \ \text{cm}^2\)
- Chord length = \(8 \ \text{cm}\)
- Triangle area = \(16\sqrt{3} \ \text{cm}^2\)