A Triangular Field

A farmer has a three-sided field and wants the angle at one corner, to set a fence post. Pacing out the three sides is easy; measuring the angle is not.

Sides 80, 100 and 130 m, and the angle φ at P between the 100 m and 130 m sides, across from 80 m: ${80}^{2} = {100}^{2}+{130}^{2}-2\cdot 100\cdot 130\cdot \cos\left(\varphi\right)$, so cos φ = 20500/26000 and φ is about 37.96°.

This is the smallest angle of the field, since 80 m is the shortest side. A good check: the angles at the other corners must be larger.