Solving triangles – measuring a river's width with the Laws of Sines and Cosines

What it shows

Solving a triangle means finding its unknown sides and angles from three known parts, using the Law of Cosines, the Law of Sines and the angle sum of 180°. Surveyors use this to measure distances they cannot walk: from two stakes A and B on one bank they measure the baseline AB and the angles to a tree C across the river, then compute AC and the width d = AC·sinA. The model assumes flat ground, straight parallel banks and an instrument that reads both angles too high by the same amount. Given two sides and a non-included angle, there may be no triangle, one or two.

How to use

Choose a Problem. In Measure a river's width, drag stakes A, B and tree C, and change True river width and Angle-measuring error to see how the error in d depends on the baseline. In the other problems drag the vertices: known parts are orange, calculated parts blue. In Two sides and a non-included angle, drag C and P to get 0, 1 or 2 triangles. Reset restores the start.

Parameters you can change

• Problem Measure a river's width, Three sides known, Two sides and the included angle, One side and two adjacent angles, Two sides and a non-included angle
• True river width 20–150 m
• Initial distance AB 20–200 m
• Angle-measuring error 0–3 °

Questions to explore

1. With a 1° angle error, does a short or a long baseline AB give a more accurate river width?
2. Given a, b and an acute angle A, when do two different triangles fit the data?
3. When three sides are known, why is it wise to find the largest angle with the Law of Cosines?