Arc length: the proportion method
Tutoring math 12, you cover arc length. The math tutor describes a method students from a generation ago might appreciate.
Arc length is distance along the circumference of a circle. Following a 360° angle, you of course travel the entire circumference, which is 2πr, r being the radius.
What is you’re traveling less the 360°? How can you calculate the corresponding arc length?
Example: Calculate the arc length of a 110° angle on a circle of diameter 15 cm.
Solution: First, realize that half the diameter is the radius. Therefore, the radius of this circle is 7.5cm.
Next, set up the following proportion:
Next, we invoke the old “cross multiplication” trick described here. It yields
Dividing both sides by 360, we get
Apparently the arc length, if we only traverse 110°, is 14.4cm. Given that the arc length would be 2π(7.5)=47.1cm for the entire circle, our answer makes sense. 110 is just under a third of 360; correspondingly, 14.4cm is just under a third of 47.1cm.
I’ll be covering another method for arc length soon. Cheers:)
Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.