Central angle

The formula to calculate the length of an arc of a circle is:

Arc length = $r\theta$

Where:

$r$ = radius of the circle

$\theta$ = central angle in radians

The circumference of a circle can be calculated using the formula:

Circumference = $\pi \times \text{diameter} = 2\pi \times \text{radius}$

Where $\pi \approx 3.141592...$

Radians are the standard mathematical way to measure angles. One radian is equal to the angle created by taking the radius of a circle and stretching it along the edge of the circle. In simpler terms, one radian is the angle formed when the length of the arc is equal to the radius of the circle.

The radian is a pure mathematical measurement and, therefore, is preferred by mathematicians over degree measures. For everyday work, the degree is easier to work with, but for mathematical analysis, the radian provides more precise results.

While radians are commonly used in mathematics and physics, you might not often encounter them in areas like construction or surveying, where degrees are more commonly applied.

The unit used to describe the measurement of an angle that is most familiar is the degree. To convert radians to degrees or degrees to radians, the following relationships are used:

Angle in degrees = angle in radians × $(180/\pi)$

Angle in radians = angle in degrees × $(\pi/180)$

This means that:

180 degrees = $\pi$ radians

360 degrees = $2\pi$ radians

Example 1: Convert 45 degrees to radians

Given that we need to convert 45 degrees to radians:

Using the formula:

radians = degrees × $(\pi/180)$

radians = $45 \times (\pi/180)$

radians = $\pi/4$

So, 45 degrees is equal to $\pi/4$ radians or approximately 0.785 radians.

Example 2: Convert $\pi/3$ radians to degrees

Now let's convert $\pi/3$ radians to degrees:

Using the formula:

degrees = radians × $(180/\pi)$

degrees = $(\pi/3) \times (180/\pi)$

degrees = $180/3 = 60°$

So, $\pi/3$ radians is equal to 60 degrees.