radians - das Bogenmaß
In a circle, another way to measure the turn an angle makes is by regarding the arc length in proportion to the circumference of the full circle.
One radian is the angle subtended by the arc equal to the radius of a circle.
We can compare the angles in degrees and in radians as follows.
$${\frac{\alpha}{360°} = \frac{l}{2\pi r }}$$
Where l is the arc subtended by the angle $\alpha$ and r is the radius.
Let x be the angle in radians and ${ x = \frac{l}{r}}$ or ${ l = xr}$
where l is the arc length and r is the radius. Then,
$${\frac{\alpha}{360°} = \frac{x}{2\pi}}$$
$${x = 2 \pi \cdot \frac{\alpha}{360°}}$$
common angles used in trigonometry
| Degrees | Radians |
|---|---|
| 0° | 0 |
| 15° | ${\frac{\pi}{12}}$ |
| 30° | ${\frac{\pi}{6}}$ |
| 45° | ${\frac{\pi}{4}}$ |
| 60° | ${\frac{\pi}{3}}$ |
| 90° | ${\frac{\pi}{2}}$ |
| 120° | ${\frac{2\pi}{3}}$ |
| 135° | ${\frac{3\pi}{4}}$ |
| 150° | ${\frac{5\pi}{6}}$ |
| 180° | ${\pi}$ |
| 270° | ${\frac{3\pi}{2}}$ |
| 360° | ${2\pi}$ |
$\frac{180°}{\pi} \approx 57.295779513°$ |
1 |
the area of a sector
The area of a circle measured in degrees is
$${A_{circle} = \pi r^2}$$
A sector is a fraction of the circle equivalent to the arc length l to circumference.
So, the area is that fraction.
$${A_{sector} = \frac{l}{2 \pi r} \pi r^2 = \frac{1}{2} lr}$$
See also a page on the lateral surface of a cone
special case - unit circle
In a unit circle, the radius is 1.
The arc length l of a sector is equivalent to the angle x in radians.
The area of the sector in a unit circle is
$${A_{sector} = \frac{1}{2} l = \frac{1}{2} x}$$
where x is the angle of the sector in radians.