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

Radians Defining the radian thumb|The radian is defined as an angle subtended by an arc the same length as the radius. A 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 Degrees 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 Arc length thumb|The length L is the arc. The area in green is the sector. 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 The 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 . Coordinate Geometry Trigonometry