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:

110/360=x/(2π7.5)

Next, we invoke the old “cross multiplication” trick described here. It yields

360x=110(2π7.5)

Dividing both sides by 360, we get

x=110(2π7.5)/360=14.4cm

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.

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