TwiddleLab / Math / Shapes and Proof
Similar Triangles and Shadows
What you will learn
- Similar triangles have the same angles; every side is multiplied by the same scale factor.
- At one moment the sun makes similar triangles: height ÷ shadow is the same for everything.
- So an unknown height = known height × scale factor.
How the lesson goes
How tall is that tree? You have a 1 m stick and a sunny day.
Bet first. Drag the lever.
Move the sun. Watch both shadows.
Drag the sun across the sky. Three different spots get logged in the table.
Blow up the stick's triangle.
Stick, shadow and sun ray make a small triangle. Slide the scale until it lies exactly on the tree's triangle.
Both triangles have a right angle at the ground and the same sun angle, so all their angles match. Matching angles mean one is the other scaled up.
Same shape, different size.
Triangles with the same angles are similar: one is a scaled copy of the other.
Every side is multiplied by the same scale factor. Here it is 8: shadow 1.5 m becomes 12 m, height 1 m becomes 8 m.
So ratios stay the same: height ÷ shadow is equal for the stick and the tree.
Later in the day, a 1 m stick casts a 1.5 m shadow.
Set your own height. Then drag the tip of your shadow to where it falls.
Same sun, so your triangle is similar to the stick's: scale factor = your height ÷ 1 m, and the shadow scales by the same factor.
Game: height sniper.
Same sun. Each shadow is measured. Set the height you work out, then Measure. Up to 100 points a target.
An AI that reads digits shrinks every picture to 28 × 28 pixels first.
Shrinking is fine if the shape survives. Tap every copy that is similar to the original.
A similar copy keeps all angles and all ratios of lengths. Stretching only one way changes the angles of the 7.
A 2 m post casts a 3 m shadow. At the same moment, a flagpole casts 12 m. How tall is the flagpole?
Same sun, so the triangles are similar. Scale factor = 12 ÷ 3 = 4, and height = 2 × 4 = 8 m.
Before and after.
Switch on your bet from the start and the measurement.
Previous lesson: Same Shape, Same Size · Next lesson: Pythagoras by Rearranging · Part of the course: Shapes and Proof
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