# How do we know the lengths of the sides of a triangle?

Icebreaker
Besides empirical measuring and without Pythagoras' knowledge of the theorem, how do we know the lengths of the sides of a triangle? Is this dealt with in "The Elements"?

That is, if we define the length of one side, with three angles, how do we know what's the measure of the other two sides using the most elementary methods?

Last edited by a moderator:

James R
Homework Helper
Gold Member
As far as I am aware, there are no more elementary methods for solving that kind of problem than by using trigonometry.

Suppose we are given a length L for the hypotenuse of this right triangle, and an angle a. Then the other two sides of the triangle are given by:

$$L_1 = L (a- \frac{a^3}{3!} + \frac{a^5}{5!}...)$$

$$L_2 = L (1+ \frac{a^2}{2!} - \frac{a^4}{4!}...)$$

This can be deduced from a knowledge of polynomials, and viewing the sine of the angle as a function. It would be interesting to derive these formulae by a purely geometric argument.