# A-level Mathematics/CIE/Pure Mathematics 1/Circular Measure

## Radians

edit### Defining the radian

editA **radian** is a unit for measuring angles. It is defined as the angle subtended by an arc that is as long as the radius. As a consequence of this, there are radians in a full circle, because the length of the circumference is times the length of the radius.

### Converting between radians and degrees

editDegrees are another common unit for measuring angles. There are in a circle, thus .

To convert from degrees to radians, multiply the number of degrees by .

e.g. is equal to

To convert from radians to degrees, multiply the amount of radians by .

e.g. radians is equal to

## Geometric Calculations

edit### Arc length

edit**Arc length** is, unsurprisingly, the length of a circular arc. This length depends on the size of the radius and the angle that the arc subtends.

We can think of an arc as a fraction of the circumference . This means that the arc length is the angle divided by a full circle times the length of the circumference: which can be simplified to .

e.g. The arc length for an arc with radius and angle is .

### Sector areas

editThe area of a sector can be derived in a similar way: it is a fraction of the area of a circle. The area of a circle is , so the area of a sector is .