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Mathematics
Similar triangles — Measuring height with shadows
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A vertical reference stick and a flagpole stand on level ground. Measure both shadows together to infer the pole height from corresponding sides.
Same time · level ground
● Reference stick■ Flagpole
Drag the ruler to the shadow tip
Changing sunlight changes both shadows together.
1.5 m
Using the rulers
The reference stick is already measured. Drag the ruler to the pole shadow tip, then explore how to find its height.
Pause the measurement, or drag the ruler yourself.
Arrow keys: 0.01 m; Page Up/Down: 0.1 m; Home/End: limits. Use Measure full shadow to attach exactly to the tip.
Same time · level ground
Reference stick: Attached to shadow tip. Flagpole: Segment being measured.
Measure both full shadows to calculate the height.
H / h = S / s
H = h × S / s
Pole height equals stick height multiplied by pole shadow divided by stick shadow.
△ABC ∼ △A′B′C′
Both triangles have a right angle at the base. Parallel sunlight makes equal angles at the shadow tips. The triangles are similar by angle–angle.
Model notes
- Both objects are vertical on level ground and measured at the same time. Sun rays are treated as parallel across a small area.
- The scene is a cross-section along the shadows. The Sun icon indicates a direction, not a nearby light source or an actual time of day.
- The digital rulers can attach to shadow tips. Height is calculated only when both rulers are attached; a free segment is not a shadow measurement.
- The model excludes blurred shadows, tilted sticks, slopes and instrument error. Real measurements have uncertainty.