弧度法と扇形

In a circle of radius $r$, one radian is the central angle for which the arc length equals the radius. Since a full circle has arc length $2\pi r$, we get $360^{\circ }=2\pi$ radians, i.e. the key relation $\pi$ radians $=180^{\circ }$. To convert degrees to radians multiply by $\frac{\pi }{180}$; to convert radians to degrees multiply by $\frac{180}{\pi }$. For example $90^{\circ }=\frac{\pi }{2}$ and $60^{\circ }=\frac{\pi }{3}$. Exams usually want exact answers involving $\pi$, so knowing the common angles by heart saves time.

For a sector of radius $r$ and central angle $\theta$ (in radians), the arc length is $s=r\theta$. You can derive this by taking the fraction $\frac{\theta }{2\pi }$ of the full circumference $2\pi r$: $2\pi r\times \frac{\theta }{2\pi }=r\theta$. The formula holds only when $\theta$ is in radians — substituting degrees directly is a mistake. If a question gives the angle in degrees, convert to radians first.

The area of that same sector is $A=\frac{1}{2}{r}^2\theta$. Again, taking the fraction $\frac{\theta }{2\pi }$ of the full area $\pi r^2$ gives $\pi r^2\times \frac{\theta }{2\pi }=\frac{1}{2}{r}^2\theta$. Using $s=r\theta$, this can also be written $A=\frac{1}{2}rs$, which is handy when $r$ and $s$ are known. As always, $\theta$ must be in radians.

The region bounded by a chord and an arc is a segment. Its area is the sector area minus the area of the triangle formed by the two radii and the chord. Using the two-sides-and-included-angle formula, that triangle has area $\frac{1}{2}{r}^2\sin \theta$, so the segment area is $\frac{1}{2}{r}^2\theta -\frac{1}{2}{r}^2\sin \theta =\frac{1}{2}{r}^2(\theta -\sin \theta )$. When evaluating $\sin \theta$, keep the calculator in radian mode. If asked for the perimeter of a sector, add the arc $r\theta$ to the two radii $2r$, giving $r\theta +2r$.