Circular measure

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 $\dfrac{180}{\pi}$.

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.

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

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

Write from an arc length of 6

• A circle has radius 4 and the angle is 1.5 radians.
• Find the arc length and the sector area.