Angular Degree

A circle has 360 degrees. Each degree can have 60 minutes (designated as ' ) and each minute can have 60 seconds (designated as " ). Since we can not convert minutes or seconds into radians directly we need to convert the minutes and seconds into a decimal number. Here is the formula:

Convert ${\displaystyle X^{\circ }}$ Y' Z" into degrees. ${\displaystyle X+\left(Y\times {\frac {1}{60}}\right)+\left(Z\times {\frac {1}{3600}}\right)=X+{\frac {Y}{60}}+{\frac {Z}{3600}}}$ For a practical example, convert ${\displaystyle 23^{\circ }}$ 18' 38" into degrees. ${\displaystyle 23+{\frac {18}{60}}+{\frac {38}{3600}}=23.3106}$

Angular Radian

A radian is the angle subtended at the centre of a circle by an arc of its circumference equal in length to the radius of the circle. Since we know that that the formula for the circumference of a circle is ${\displaystyle C=2\pi r}$ we can determine that there are ${\displaystyle 2\pi }$ radians in a circle. We abbreviate radians as rad.

Conversion Between Degrees and Radians

Mathematics requires us to use radians for most angular measurements. Therefore we need to know how to convert from degrees to radians Since we know that there are ${\displaystyle 360^{\circ }}$ degrees or ${\displaystyle 2\pi }$ radians in a circle. We can determine these equations: ${\displaystyle 1^{\circ }={\frac {\pi }{180}}}$

${\displaystyle 1\ radian={\frac {180}{\pi }}}$

So we can write these general formulae.

${\displaystyle X^{\circ }={\frac {X\times \pi }{180}}}$

${\displaystyle X\ radian\left(s\right)={\frac {X\times 180}{\pi }}}$

Here is an example convert ${\displaystyle 20^{\circ }}$ into radians

${\displaystyle 20^{\circ }={\frac {20\times \pi }{180}}={\frac {1}{9}}\pi \ rad}$

Convert ${\displaystyle {\frac {1}{9}}\pi }$ into degrees.

${\displaystyle {\frac {1}{9}}\pi \ radian\left(s\right)={\frac {{\frac {1}{9}}\pi \times 180}{\pi }}=20^{\circ }}$

Arc Length

In most cases it is very difficult to measure the length of an arc with a ruler. Therefore we need to use a formula in order to determine the length of the arc. The formula that we use is:

${\displaystyle Arc\ Length=(\theta \ in\ radians)(radius)\,}$ in symbols this is ${\displaystyle s=\theta r\,}$. Note: θ need to be in radians

Here is an example, determine the length of the arc created by a sector with a 6 cm radius and an angle of ${\displaystyle 53^{\circ }}$.

The first thing we need to do is convert θ from degrees to radians.

${\displaystyle 53^{\circ }={\frac {53\pi }{180}}rad}$

Now we can calcuate the length of the arc.

${\displaystyle s={\frac {53\pi }{180}}\times 6\approx 5.55cm}$

Area of a Sector

The area of a sector can be found using this formula:

${\displaystyle Area={\frac {1}{2}}r^{2}\theta }$Note: θ need to be in radians.

Calculate the area of a sector with a 3 cm radius and an angle of 2π.

${\displaystyle Area={\frac {1}{2}}3^{2}\times 2\pi \approx 28.3cm^{2}}$