Circular measure
| English | Chinese | Pinyin |
|---|---|---|
| radians | 弧度 | hú dù |
| arc length | 弧长 | hú zhǎng |
| sector area | 扇形面积 | shàn xíng miàn jī |
| segment | 弓形 | gōng xíng |
Why radians 弧度?
- Engineers measure angles in degrees. Mathematicians prefer radians — and for good reason.
- A radian is the natural unit: it's the angle that cuts off an arc equal to the radius.
- With radians, the formulas for arc length 弧长 and sector area 扇形面积 become beautifully simple.
What is a radian?
- A radian is the angle that cuts off an arc equal to the radius.
- The link: $\pi \text{ radians} = 180^\circ$.
- degrees → radians: multiply by $\dfrac{\pi}{180}$; radians → degrees: multiply by $\dfrac{180}{\pi}$.
Common conversions. $90^\circ = \dfrac{\pi}{2}$ rad; $\;60^\circ = \dfrac{\pi}{3}$ rad; $\;45^\circ = \dfrac{\pi}{4}$ rad; $\;30^\circ = \dfrac{\pi}{6}$ rad.
π radians equals how many degrees?
By definition, π radians = 180°.
60° in radians is:
60° × π/180 = 60π/180 = π/3.
Arc length and sector area
- For a sector with radius $r$ and angle $\theta$ in radians:
- arc length $s = r\theta$,
- sector area $A = \tfrac12 r^2 \theta$.
The elegance of radians: arc length $= r\theta$ and sector area $= \tfrac{1}{2}r^2\theta$. No messy $\dfrac{\theta}{360}$ fractions needed.
Radians only. The formulas $s = r\theta$ and $A = \tfrac{1}{2}r^2\theta$ work only when $\theta$ is in radians. If your angle is in degrees, convert first.
The sector
s = rθ · A = ½r²θ
Drag the angle θ (in radians) and the radius r — the arc length and sector area follow.
A sector has radius 5 and angle 2 radians. What is the arc length (s = rθ)?
s = rθ = 5 × 2 = 10.
A sector has radius 4 and angle 0.5 radians. What is its area (A = ½r²θ)?
A = ½ × 4² × 0.5 = ½ × 16 × 0.5 = 4.
The formula s = rθ works when θ is measured in degrees.
s = rθ only works when θ is in radians. Convert degrees to radians first.
Segment 弓形 area
- A segment is the region between a chord and the arc.
- Segment area = sector − triangle = $\tfrac12 r^2(\theta - \sin\theta)$.
- The triangle area is $\tfrac{1}{2}r^2\sin\theta$ (using the sine rule for area).
A tangent meets the circle once and is perpendicular to the radius there
A segment has r = 6 and θ = π/3 radians. The segment area is ½r²(θ − sin θ). Find it (2 dp).
½ × 36 × (π/3 − sin(π/3)) = 18 × (1.0472 − 0.8660) = 18 × 0.1812 = 3.26.
Worked example
- Sector: $r = 5$, $\theta = 2$ rad.
- Arc length $= 5 \times 2 = 10$.
- Sector area $= \tfrac{1}{2}(5^2)(2) = 25$.
- Segment area $= \tfrac{1}{2}(25)(2 - \sin 2) = 12.5(2 - 0.909) = 13.6$.
You've got it
- $\pi$ radians $= 180^\circ$; convert by $\times\dfrac{\pi}{180}$ or $\times\dfrac{180}{\pi}$
- arc length $s = r\theta$; sector area $A = \tfrac12 r^2\theta$ ($\theta$ in radians)
- segment $= \tfrac12 r^2(\theta - \sin\theta)$