Radian Measure of Angles
Radian measure is a way of describing angles using the radius of a circle. One radian is defined as the angle subtended at the center of a circle by an arc length equal to the radius of the circle.
Note$1\ rad$ is approximately equal to $57.3°$.
NoteThe full angle around a circle is 2π radians, which is equivalent to 360°.
The formula for converting between degrees and radians is:
$\text{angle in radians} = \frac{\text{angle in degrees} \times \pi}{180°}$
$\text{angle in degrees} = \frac{\text{angle in radians} \times 180°}{\pi}$
ExampleConvert 45° to radians: $45° \times \frac{\pi}{180°} = \frac{\pi}{4}$ radians
Convert $\frac{2\pi}{3}$ radians to degrees: $\frac{2\pi}{3} \times \frac{180°}{\pi} = 120°$
TipMemorize common angle conversions like 30° = $\frac{\pi}{6}$, 45° = $\frac{\pi}{4}$, and 60° = $\frac{\pi}{3}$.
Length of an Arc
The length of an arc is a portion of the circumference of a circle. The portion of the circumference an arc takes up is $\frac{\theta}{2\pi}$, where $\theta$ is the angle subtended by the arc in radians. This should (hopefully) be intuitive. Therefore,
$\text{Arc length} = \frac{\theta}{2\pi}2\pi r = r\theta$
Where:
- $r$ is the radius of the circle
- $\theta$ is the angle subtended by the arc in radians
Calculate the length of an arc that subtends an angle of $\frac{\pi}{3}$ radians in a circle with radius 5 cm.
$\text{Arc length} = r\theta = 5 \times \frac{\pi}{3} = \frac{5\pi}{3}$ cm ≈ 5.24 cm
Common MistakeRemember to always use radians for $\theta$ in this formula. If given an angle in degrees, convert it to radians first.
Area of a Sector
A sector is a region of a circle enclosed by two radii and an arc. The proportion of the area a sector takes up is also $\frac{\theta}{2\pi}$, where $\theta$ is the angle subtended by the sector in radians. Thus,
$\text{Sector area} = \frac{\theta}{2\pi}\pi r^2 = \frac{1}{2}r^2\theta$
Where:
- $r$ is the radius of the circle
- $\theta$ is the angle subtended by the sector in radians
Calculate the area of a sector with central angle $\frac{3\pi}{4}$ radians in a circle with radius 8 cm.
$\text{Sector area} = \frac{1}{2}r^2\theta = \frac{1}{2} \times 8^2 \times \frac{3\pi}{4} = 24\pi$ cm² ≈ 75.4 cm²
Relationship Between Arc Length and Sector Area
There's a neat relationship between arc length and sector area that can be useful to remember.
This relationship can be seen by comparing the formulas:
- Arc length: $r\theta$
- Sector area: $\frac{1}{2}r^2\theta = \frac{1}{2}r(r\theta)$
You can substitute arc length into the formula for sector area, which gives you the relationship:
$\text{Sector area} = \frac{1}{2} \times \text{radius} \times \text{arc length}$
TipThis relationship can be helpful in problem-solving. If you know the arc length and radius, you can quickly find the sector area without needing to calculate the angle in radians.
Expressing Answers
It's important to express answers appropriately in your exams so you don't get docked marks. When working with radians and π:
- Exact answers should be given in terms of π when possible. For example, $\frac{2\pi}{3}$ rather than 2.0944. (Sorry, engineers.)
- When a decimal approximation is required, use an appropriate number of decimal places or significant figures as specified in the question. If unspecified, always go with 3 significant figures. This is written on the front of your exam paper.
Express the area of a sector with radius 10 cm and central angle 40°:
- Convert angle to radians: $40° \times \frac{\pi}{180°} = \frac{2\pi}{9}$ radians
- Calculate area: $\text{Area} = \frac{1}{2}r^2\theta = \frac{1}{2} \times 10^2 \times \frac{2\pi}{9} = \frac{100\pi}{9}$ cm²
- Decimal approximation (to 2 d.p.): 34.91 cm²
Always include units in your final answer when working with real-world problems involving lengths or areas.
