Similar triangles: measuring with shadows
Use a stick and two shadows to find the height of a tree, a pole or a temple, whatever the height of the sun.
Core concept
Triangles with the same angles are similar: their sides are in the same ratio, the scale factor
Topic: Similarity
On a sunny afternoon a student stands a 1.5 m stick upright and measures its shadow, then measures the shadow of a tall tree nearby. The sun's rays reach both at the same angle: The amount of turn between two lines that meet, measured in degrees (°). All key words, so the stick, its shadow and the ray make a triangle with exactly the same angles as the tree, its shadow and the ray. Such triangles are similar: one is an enlargement of the other, and every side is multiplied by the same scale factor: A number that divides another exactly. 3 is a factor of 12. All key words. So tree height ÷ stick height = tree shadow ÷ stick shadow, and the tree's height can be found without climbing it.
What you will learn
- Recognise similar triangles from equal angles.
- Find a scale factor from a pair of matching sides.
- Use proportion to find an unknown length.
Before you start
- Ratio and proportion; angles of a triangle.
Materials
- A straight stick of known length
Equipment
- A measuring tape
Procedure
- Drag the sun up or down to change its angle; tap the chips to change the stick or the object.
- Read both shadows and the shadow ratio (the scale factor).
- Find the tree's height when the sun is at 45°, and record the same tree at three different sun angles.
- Measure the temple, and use the 2 m stick.
How this simulation is simplified
- The ground is level and the sun's rays are parallel.