Previous Math Race

Exercice 1, 2015-2016 first edition

Exercise 1, 2015-2016 first edition

Last year the Twente Math Relay Race took place, a competition in which secondary school students try to beat each other at solving as many mathematical exercises as possible in a certain time. Eric competed in a group together with three classmates. One of them celebrated his birthday by bringing a huge cake to school. Eric watched how the cake was cut into slices, while going over mathematical challenges in his head. He started wondering what would happen if you could keep cutting the cake into an infinite number of slices. Could he use this observation to derive the famous formula for computing the area of a circle?

Exercise A

Apparently, Eric’s assumption is correct. Derive a circle with radius R in triangles, in the same way as in the picture above. Give the formula for the area of the sum of these triangles as a function of the number of triangles N.

Exercise B

Show that the area of the circle is found when the number of triangles N goes to infinity.

Exercise C

Show in a way similar to the one above that the circumference of a circle is equal to 2πr.

Hint

The function sin(x) can be approximated using a Taylor approximation around the point x=0: