Trigonometry — Build a Clinometer and Measure a Tree's Height

Ancient astronomers invented trigonometry to turn the ANGLES they measured into the LENGTHS of triangles. You will use it to measure the height of something too tall to reach.

Tape a drinking straw along the straight edge of a protractor to sight through. Tie a short string to the centre of the protractor with a small weight on the end, so it hangs down and shows the angle. That is a clinometer -- it measures how steeply you are looking up.

Stand back from a tree or building and look through the straw at its very top. Read the angle the hanging string marks on the protractor (this tells you the angle above horizontal). Then measure your distance from the base with a tape measure. Write down the angle and the distance.

What your clinometer teaches. (1) In a right triangle the sine, cosine and tangent are ratios of the sides that depend ONLY on the angle, so a single table of them works for every triangle -- Ptolemy tabulated them around 150 AD, and Indian mathematicians shaped them into the 'sine' we use (the word comes from Sanskrit jya). (2) The tangent turns an angle you can see into a length you cannot reach -- the everyday tool of surveyors and sailors. (3) As an angle turns full circle, sine and cosine trace smooth repeating WAVES. (4) Those sine waves describe sound, light, radio, tides and alternating current, and underlie GPS, music synthesis and the Fourier analysis that compresses every image and song on your phone.