High School: Geometry
5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
It's easy for students to memorize formulas and in fact, many of them have no problem with that. Formulas are their ticket to the Math Train, which gets them from algebra to geometry and eventually, to calculus. They continue plugging and chugging their way along without a care in the world until they hit a blockade: Derivation Station.
Uh oh. All they've done up until now is memorize. How ever will they get to Calculus City?
They shouldn't worry their little heads. Derivation Station is just about taking formulas we already know, applying them logically, and coming up with a new formula. In this case, it's about arc length. (Tip: make sure students know this is not the same as arc measurement!)
Students should first draw a circle with radius r and a central angle of θ. Visually, it should be clear that the arc length s depends on both the central angle and the radius. If we look at the circumference C as one bit arc length, we can see that its central angle is 360° (or 2π radians). If C = 2πr, then the arc length should equal the central angle (in radians) times the radius, or s = θr.
If students aren't as familiar with radians, make sure to take some time and explain it to them. It can be tricky going from degrees to radians, but you can use the circumference equation to translate from one to another. (We have 2π as our factor but it must come from 360° somehow. If we say 360k = 2π and solve for k, we have our conversion factor from degrees to radians. Yippee.)
Students should also be able to derive the formula for the sector of a circle. Just like we used the C = 2πr to find arc length, we can use A = πr2 to find the area of a sector. This time, the 360 degrees translates to π, not 2π. That means we're dividing the angle by 2, so our formula should be A = ½θr2.
The key thing to note here is that θ must be in radians. You can explain to them that a radian is about 57.3°, that it's the angle at which the ratio of the arc length to the radius is 1:1, and that it's an Australian experimental music band. That last one is optional, by the way.
Make sure the entire concept of radians isn't lost on your students. If you sense that radians confuse them more than they should, have them measure the radius of a circle and then mark an arc of the same length. Students can then measure the central angles intercepting that arc with their protractors. Even if they do this for circles of various sizes, the angle measure shouldn't change. That's a radian.