Arc Length and Sector Area
Formulas
The arc length AB subtends an angle \( \theta \) is and often labelled by the letter l. The animation shows the arc length for different values of \( \theta \) for a sector with radius r. When \( \theta=360 \) the arc length is actually the circumference of a circle, hence \( s=2 \pi r \). The circumference of the circle is multiplied by the ratio \( \theta / 360 \) to calculate l: $$l = {{\theta \over 360} 2 \pi r } $$
This simplifies to: $$l = {{\theta \over 180} \pi r } $$ The area of the sector (yellow shaded region) A is the area of a circle with radius r multiplied by the fraction \( \theta / 360 \): $$A = {{\theta \over 360} \pi r^{2} } $$ The formulas are for \( \theta \) in degrees. The conversion formula from angle in degrees \( \theta_{d} \) to an angle \( \theta_{r} \) in radians $$ \theta_{d}={180 \over \pi} \theta_{r} $$ can be substituted in the formulas for the arc length and sector to obtain: $$ l= \theta r $$ $$ A={1 \over 2} \theta r^{2} $$ The conversion to radians acts as a vacuum cleaner for the various constants and simplifies the formulas.Exercise 1: Calculate angle given the arc length
Try the question on paper before seeing the solution
Solution
Starting with the formula for the arc length for angles in degrees: $$l = {{\theta \over 180} \pi r } $$ multiplying both sides by 180: $$ 180 l= \theta \pi r$$ Dividing both sides by \( \pi r \): $$ \theta= {180l\over{\pi r}} $$ substituting and we get \( \theta \) degrees.
Starting with the radians version of the arc length formula \( l=\theta r \), we just divide by r: $$ \theta={l \over{r}} $$, and substituting the above values for l and r we get \( \theta \)= radians (rounded to 2 decimal places).
Discussion
The formulas for arc length and sector area when using radians are very elegant and easy to use compared to the formulas where the angle is in degrees. Questions could involve finding the angle of a sector given the radius and area. The formulas will then have to be manipulated as follows. Starting with formula for the area in radians: $$ A={1 \over 2} \theta r^{2} $$ multiplying both sides by 2 we get: $$ 2A= \theta r^{2}$$ diving by \( r^{2} \) we get the formula: $$ \theta={{2A}\over{r^{2}}} $$ Rearranging the degrees version of the formula is very similar except there are more constants.