What it shows
A small mirror on the ground, a tape and your own eye height are enough to find the height of a pole, tree or building without climbing it. Light reflecting from the mirror forms two similar right triangles, so the heights are in proportion to the distances.
The science
By the law of reflection the angle of incidence equals the angle of reflection. If you see the top of the tree in a mirror on the ground, the ray from the top to the mirror and the ray from the mirror to your eye make equal angles with the ground. Then the right triangle formed by you and the mirror is similar to the triangle formed by the tree and the mirror, so tree height divided by your eye height is equal to the tree-to-mirror distance divided by the mirror-to-you distance.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A small flat mirror, a pocket mirror or a steel plate
- A tape measure or rope of 30 m marked in metres
- Chalk for marking the ground
- A flat open place near a tree or building
- A notebook and a calculator
- A helper
Safety
For this project: Stay on level ground, keep away from roads, wires and the edge of any roof, and never climb anything to measure. An adult must be present.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Stand a few metres from the tree on level ground and place the mirror flat on the ground in a straight line between you and the tree.
Move back, away from the mirror along the same line, until you can just see the top of the tree in the mirror, and mark the mirror spot with chalk.
Measure your eye height E from the ground while standing in the same position, with the helper using the tape. Measure with shoes on.
Measure the distance d1 from your feet to the mirror and the distance d2 from the mirror to the base of the tree.
Calculate the height of the tree: H = E times d2 divided by d1.
Repeat from two other positions on the other side of the mirror and take the average of the results. A common mistake is a tilted mirror, so check that it is flat.
Compare with a shadow-method estimate by measuring a metre stick’s shadow and the tree’s shadow at the same time of day.
Draw the triangles to scale, label the equal angles and show the calculation in your report.
What you should see
For E = 1.4 m, d1 = 2 m and d2 = 20 m the height is 14 m, and the shadow method gives an answer within about 1 m of it.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1Why are the two triangles similar?
Both are right-angled and have equal angles at the mirror, by the law of reflection.
2What does similar mean?
Same shape, equal angles and sides in the same ratio.
3Why must the mirror be flat on the ground?
A tilted mirror changes the angles of reflection.
4What is a second method for finding the height?
Measure shadows of the tree and a stick of known height at the same time and use proportion.
Take it further
Measure a building and then check against the height printed in its plan or the number of floors times 3 m.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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