Resources | Subject Notes | Mathematics
In Euclidean geometry, angles are typically measured in degrees. However, in mathematics, especially in calculus and physics, angles are often expressed in radians. A radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
The relationship between degrees ($\degree$) and radians (rad) is given by:
$$ \text{radians} = \frac{\pi}{180} \times \text{degrees} $$
Conversely:
$$ \text{degrees} = \frac{180}{\pi} \times \text{radians} $$
For example, 90 degrees is equal to $$\frac{\pi}{2}$$ radians.
The arc length ($s$) of a circle is the distance along the circumference subtended by a given angle ($\theta$) at the center of the circle. The radius of the circle is denoted by $r$.
The formula for arc length is:
$$ s = r\theta $$
where $\theta$ is measured in radians.
If $\theta$ is given in degrees, we first convert it to radians using the formula above before applying this formula.
A sector of a circle is a region bounded by two radii and an arc. The area of a sector ($A$) is a fraction of the total area of the circle. This fraction is determined by the angle subtended at the center of the circle ($\theta$) in radians.
The formula for the area of a sector is:
$$ A = \frac{1}{2}r^2\theta $$
where $r$ is the radius and $\theta$ is the angle in radians.
If $\theta$ is given in degrees, we first convert it to radians before using this formula.
For small angles (typically less than 10-15 degrees), we can use the approximation that the arc length of a circle is approximately equal to the length of the chord subtended by the angle, and the area of a sector is approximately equal to half the square of the angle.
The approximation for arc length is:
$$ s \approx \frac{\theta}{2r} $$
where $\theta$ is in radians.
The approximation for the area of a sector is:
$$ A \approx \frac{1}{2} \theta^2 $$
where $\theta$ is in radians.
These approximations are useful in situations where the angle is small and calculations are simplified.
| Concept | Formula | Units |
|---|---|---|
| Radian to Degree | $$ \text{degrees} = \frac{180}{\pi} \times \text{radians} $$ | Degrees ($\degree$) |
| Degree to Radian | $$ \text{radians} = \frac{\pi}{180} \times \text{degrees} $$ | Radians (rad) |
| Arc Length | $$ s = r\theta $$ | Length |
| Area of Sector | $$ A = \frac{1}{2}r^2\theta $$ | Area |
| Arc Length (Small Angle Approx.) | $$ s \approx \frac{\theta}{2r} $$ | Length |
| Area of Sector (Small Angle Approx.) | $$ A \approx \frac{1}{2} \theta^2 $$ | Area |