Edexcel IAL - Pure Maths 1- 3.2 Radian Measure, Arc Length, and Sector Area- Study notes - New syllabus
Edexcel IAL – Pure Maths 1- 3.2 Radian Measure, Arc Length, and Sector Area -Study notes- New syllabus
Edexcel IAL – Pure Maths 1- 3.2 Radian Measure, Arc Length, and Sector Area -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- Radian Measure & Converting Between Units
- Arc Length
- Sector Area
Radian Measure & Converting Between Units
Angles can be measured in degrees or radians. Radians are the natural unit used in advanced mathematics, calculus and trigonometry because they relate directly to arc length.
Definition
One radian is the angle formed at the centre of a circle by an arc whose length equals the radius.
If arc length is \( s \) and radius is \( r \), then
\( s = r\theta \) where \( \theta \) is in radians.
Key Relations & Conversions
| Concept | Formula |
| Degrees to Radians | \( \theta_{\text{rad}} = \dfrac{\pi}{180}\theta_{\deg} \) |
| Radians to Degrees | \( \theta_{\deg} = \theta_{\text{rad}} \cdot \dfrac{180}{\pi} \) |
| Full Rotation | \( 360^\circ = 2\pi \text{ rad} \) |
| Half Rotation | \( 180^\circ = \pi \text{ rad} \) |
| Quarter Rotation | \( 90^\circ = \dfrac{\pi}{2} \text{ rad} \) |
Notes
- Radians must be used in formulas like \( s = r\theta \) or \( A = \dfrac12 r^2\theta \).
- Always convert degrees to radians before using trigonometric formulae involving arc length or sector area.
- The radian measure gives more natural behaviour in calculus.
Example
Convert \( 150^\circ \) to radians.
▶️ Answer / Explanation
\( 150^\circ = 150 \cdot \dfrac{\pi}{180} = \dfrac{5\pi}{6} \)
Example
Convert \( 2.8 \) radians into degrees.
▶️ Answer / Explanation
\( \theta_{\deg} = 2.8 \cdot \dfrac{180}{\pi} \)
The exact answer remains in this form unless decimal evaluation is needed.
Example
An angle \( \theta \) satisfies \( 5\theta_{\text{rad}} = 300^\circ \). Find \( \theta \) in radians.
▶️ Answer / Explanation
Convert degrees to radians:
\( 300^\circ = 300 \cdot \dfrac{\pi}{180} = \dfrac{5\pi}{3} \)
Equation:
\( 5\theta = \dfrac{5\pi}{3} \)
\( \theta = \dfrac{\pi}{3} \)
Arc Length
The arc length of a curve is the distance measured along the curve itself. For a circle, the arc length of a sector depends on the radius and the angle (in radians) subtended at the centre.
In a circle of radius \( r \), if the angle at the centre is \( \theta \) radians, then the arc length is directly proportional to both.
Key Formulae
| Concept | Formula |
| Arc Length | \( s = r\theta \) (\( \theta \) must be in radians) |
| Full Circle | Total circumference \( = 2\pi r \) |
| Fraction of a Circle | If angle \( \theta \) corresponds to fraction \( \dfrac{\theta}{2\pi} \) of the circle Arc length \( = \dfrac{\theta}{2\pi} \times 2\pi r = r\theta \) |
Notes
- \( \theta \) must always be converted to radians before using \( s = r\theta \).
- The formula is derived from the proportionality: \( \dfrac{s}{2\pi r} = \dfrac{\theta}{2\pi} \).
- If \( \theta \) is given in degrees, convert using \( \theta_{\text{rad}} = \dfrac{\pi}{180}\theta_{\text{deg}} \).
Example
Find the arc length for a sector of radius \( r = 6 \) and central angle \( \theta = \dfrac{\pi}{4} \).
▶️ Answer / Explanation
Use formula:
\( s = r\theta = 6 \cdot \dfrac{\pi}{4} = \dfrac{6\pi}{4} = \dfrac{3\pi}{2} \)
Example
A wheel of radius \( 20 \text{ cm} \) rotates through \( 72^\circ \). Find the arc length travelled on its rim.
▶️ Answer / Explanation
Convert angle to radians:
\( \theta = 72^\circ \cdot \dfrac{\pi}{180^\circ} = \dfrac{2\pi}{5} \)
Arc length:
\( s = r\theta = 20 \cdot \dfrac{2\pi}{5} = 8\pi \text{ cm} \)
Example
A circular track has radius \( 50 \text{ m} \). A runner completes an arc length of \( 120 \text{ m} \). Find the angle (in degrees) subtended at the centre.
▶️ Answer / Explanation
Use \( s = r\theta \):
\( 120 = 50\theta \)
\( \theta = \dfrac{120}{50} = 2.4 \text{ radians} \)
Convert to degrees:
\( \theta_{\deg} = 2.4 \cdot \dfrac{180}{\pi} \)
Area of a Sector
A sector is the region of a circle enclosed by two radii and the arc between them. The area depends on the radius of the circle and the angle at the centre (in radians).
In a circle of radius \( r \), if the central angle is \( \theta \) radians, the sector area is a fraction of the full circle.
Key Formulae
| Concept | Formula |
| Area of Sector | \( A = \dfrac12 r^2\theta \) (\( \theta \) in radians) |
| Full Circle Area | \( \pi r^2 \) |
| Fraction of a Circle | Fraction \( = \dfrac{\theta}{2\pi} \) Sector area \( = \dfrac{\theta}{2\pi} \cdot \pi r^2 = \dfrac12 r^2\theta \) |
Notes
- \( \theta \) must be in radians when using the formula.
- The formula comes from proportionality with the full circle area.
- If \( \theta \) is given in degrees, convert using \( \theta_{\text{rad}} = \dfrac{\pi}{180}\theta_{\text{deg}} \).
Example
Find the area of a sector with radius \( r = 4 \) and central angle \( \theta = \dfrac{\pi}{3} \).
▶️ Answer / Explanation
Use area formula:
\( A = \dfrac12 r^2\theta \)
\( A = \dfrac12 (4^2)\cdot \dfrac{\pi}{3} = 8 \cdot \dfrac{\pi}{3} = \dfrac{8\pi}{3} \)
Example
A sector has radius \( 12 \text{ cm} \) and area \( 18\pi \text{ cm}^2 \). Find the angle \( \theta \) in radians.
▶️ Answer / Explanation
Use formula:
\( 18\pi = \dfrac12(12^2)\theta \)
\( 18\pi = 72\theta \)
\( \theta = \dfrac{18\pi}{72} = \dfrac{\pi}{4} \)
Example
A circular slice has area \( 30 \text{ cm}^2 \) and central angle \( 1.2 \) radians. Find the radius of the circle.
▶️ Answer / Explanation
\( A = \dfrac12 r^2\theta \)
\( 30 = \dfrac12 r^2(1.2) \)
\( 30 = 0.6 r^2 \)
\( r^2 = \dfrac{30}{0.6} = 50 \)
\( r = \sqrt{50} \)