TL;DR:
Similar triangles have equal angles and sides in the same ratio, the scale factor.
Parallel sunlight makes a stick and a tree form similar triangles with their shadows.
Match corresponding sides, set up one proportion, and check that the answer is reasonable.
You cannot climb a tree with a tape measure, but you can measure its shadow. If a 1.5 m stick casts a 2 m shadow beside it, and the tree's shadow is 12 m, how tall is the tree? The answer needs only one idea: two triangles with the same shape.
Same shape, different size
Two triangles are similar if their angles match one for one. When that happens, the sides are in proportion: each side of the larger triangle is the same multiple of the matching side of the smaller one. That multiple is the scale factor. A 3-4-5 triangle and a 6-8-10 triangle are similar with scale factor 2.
A useful fact: two angles are enough. If two angles of one triangle equal two angles of another, the third angles must match too, because the angles in a triangle always add to 180°.
Why the stick and the tree make similar triangles
Sunlight arrives almost parallel at one place and time. So the sun's rays make the same angle with the ground for the stick as for the tree. The stick and the tree both stand vertically, so each triangle has a right angle at the base. Two matching angles are enough, so the triangles are similar.
Worked example: a 1.5 m stick with a 2 m shadow, and a tree with a 12 m shadow
Sketch both triangles. Small: height 1.5, shadow 2. Large: height
h, shadow 12.Match corresponding sides: height to height, shadow to shadow.
Write the proportion:
h / 12 = 1.5 / 2.Multiply both sides by 12:
h = 12 × 1.5 / 2 = 18 / 2 = 9.
The tree is 9 m tall.
Check that it makes sense
Scale factor. The shadow is
12 / 2 = 6times longer, so the height should be 6 times longer:1.5 × 6 = 9. Same answer.Ratio of height to shadow. For the stick it is
1.5 / 2 = 0.75. For the tree,9 / 12 = 0.75. Equal ratios are the signature of similar triangles.Sizes. A tree taller than a stick should have a longer shadow. It does.
Angle. The sun's angle satisfies
tan θ = 0.75, soθ ≈ 36.9°, the same for both.
Another way to set it up
You can also compare the two triangles by their own sides instead of across. The stick is 1.5 and its shadow is 2, so the stick is 1.5 / 2 = 0.75 of its shadow. Apply that to the tree: 0.75 × 12 = 9. Both setups give the same answer because they rearrange the same equation. Choose the one whose arithmetic you find easier.
Using the idea with a mirror or a pond
If shadows are not available, a small mirror on flat ground works. You stand where you see the top of the tree in the mirror. The ray reflects at equal angles, so you and the tree form similar right triangles. Say your eye is 1.6 m above the ground, you are 2 m from the mirror, and the mirror is 10 m from the tree. Then the tree is 1.6 × 10 / 2 = 8 m tall. Check the scale factor: 10 / 2 = 5, and 1.6 × 5 = 8.
Try one yourself: a flagpole
A person 1.8 m tall casts a 2.4 m shadow. At the same moment a flagpole casts a 10 m shadow. Set up h / 10 = 1.8 / 2.4. The ratio 1.8 / 2.4 = 0.75, so h = 10 × 0.75 = 7.5 m. Check with the scale factor: 10 / 2.4 ≈ 4.17, and 1.8 × 4.17 ≈ 7.5.
Missing sides in any pair of similar triangles
Shadows are one use, but the same method works on any similar pair. Triangle A has sides 4, 6, 8. Triangle B is similar, and its shortest side is 10. The scale factor is 10 / 4 = 2.5, so the other sides are 6 × 2.5 = 15 and 8 × 2.5 = 20. Check the ratios: 4/10 = 6/15 = 8/20 = 0.4. Always pair the shortest with the shortest, and the longest with the longest, when the angles are not given.
Common mistakes
Matching sides that do not correspond, such as comparing the stick's height to the tree's shadow.
Measuring the shadows at different times of day, which changes the sun's angle.
Mixing units, such as a stick in centimetres and a shadow in metres.
Leaning the stick, or standing on a slope, so the triangles are no longer right-angled.
Adding the same amount to both sides instead of multiplying by the scale factor. The triangles grow by a factor, not by a fixed length.
Where you will use it next
Scale factors link to ratios and proportions with maps and recipes, where the same reasoning shrinks a landscape onto a page. Right triangles with a fixed angle always have fixed side ratios, which is the starting point for SOH CAH TOA and when to use each ratio. The sizes of the squares on the sides come from the Pythagorean theorem.
Frequently asked questions
Do similar triangles have equal areas?
No. If the scale factor is k, lengths scale by k but areas scale by k². For factor 2, a 3-4-5 triangle has area 6 and the 6-8-10 triangle has area 24, which is 6 × 4.
Does this work on a cloudy day?
Not if there is no clear shadow. You need a sharp shadow edge to measure its length, and a flat patch of ground.
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