Teaching Equation of Circle

Question

I am teaching equation of circle for high school students. They asked me two questions:

1. Why do students have to study equation of circle although they studied the plane version (without $Oxy$ coordinates)?

2. What is the real life applications of circle equation?

I tried to find the anwsers, but I just found some application of circle in general. Can anyone show me something interesting?

Thanks for your helps.

Answer 1

Find a radius of the circle, which touched to a graph of the function $y=x^4$ in three points.

How we can solve this problem without an equation of the circle?

Answer 2

You are sitting in your office and a client walks in with a order to manufacture a circular/round table for a square or rectangle hall in his mansion.

He says make me a biggest table that I can place at one of the corners of my hall so that half of the hall is empty/vacant.

What seems smart way of doing it?

Going to the mansion and measuring the dimensions of the hall or asking the architect or the owner for the floor plan of the hall that details the dimensions of the hall?

Is this example practical enough?

Answer 3

1. Why do students have to study equation of line although they studied the plane version? If you have an answer for this last question, you should have an answer for the former. There is nothing special about the object. The key question is: why to introduce a system of coordinates in the geometry? I think you can get a satisfactory answer by searching for the history and benefits of analytic geometry. The essential idea is to merge algebra and geometry. Here are two quotes in this line:

"Few academic experiences can be more thrilling to the student of advanced high school or beginning college mathematics than an introduction to this new and powerful method of attacking geometrical problems. The task of establishing a theorem in geometry is cleverly shifted to that of establishing a corresponding theorem in algebra." (Historical Topics for the Mathematics Classroom, p. 180).

"Ever since then algebra and geometry have worked together, to the advantage of both. The concept of coordinates was the first really fundamental contribution to geometry after the Greeks." (Elementary Geometry from an Advanced Standpoint, p. 249).

2. You can use it to calculate an estimation for the radius of the Earth (assuming that it is a sphere). For this, go to a large enough lake with a tape measure, a boat, a long-range laser and a helicopter, and perform the experiment described in this video. Using the data given in the video (in meters) and applying the circle equation $x^2+y^2=r^2$ for a suitable coordinate system, you conclude that the radius $R$ of the Earth satisfies $$x_1^2+(R-1.82)^2=R^2\qquad\text{and}\qquad x_2^2+(R-7.31)^2=R^2$$ for $$x_1\cong 4328\qquad\text{and}\qquad x_2\cong 9656.$$ Thus $R\cong 6785$ kilometers, which is a relatively good approximation of the real value.