Similarity

Solving with Similar Triangles

Using similar triangles to find missing side lengths and heights.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Similarity Criteria

Measuring What You Can't Reach

Before reading on, think about how you might find the height of a tall tree without climbing it, without a ladder, and without any specialized measuring tool — using only a tape measure, sunlight, and something you already know your own height to be.

On a sunny day, both the tree and a person standing nearby cast shadows at the exact same time, from the exact same sun angle. The tree, its shadow, and the sun's rays form a large right triangle; you, your shadow, and the same sun rays form a smaller right triangle with the identical angle where the ground meets the light. By AA similarity, those two triangles are similar — and similar triangles' corresponding sides share one common scale factor, which turns an unreachable height into a solvable proportion using only measurements taken safely on the ground.

Worked Example — Solving an Indirect Measurement Problem

A person 5.5 feet tall casts a 4-foot shadow at the same moment a tree casts a 28-foot shadow. Find the tree's height. Set up a proportion between corresponding sides of the similar triangles: (person's height)/(person's shadow) = (tree's height)/(tree's shadow), so 5.5/4 = h/28. Cross-multiply: 4h = 154, giving h = 38.5 feet.

Worked Example — Solving for a Missing Side in Overlapping Similar Triangles

Two triangles share a vertex and have parallel bases, making them similar by AA (the shared vertex angle, plus congruent corresponding angles from the parallel sides). The larger triangle's base is 15, and its side from the shared vertex to the base is 12. The smaller triangle's corresponding side from the shared vertex is 8. Find the smaller triangle's base. Set up the proportion using corresponding sides: base/15 = 8/12, so base = 15 × (8/12) = 10.

Tip

Before setting up any proportion between similar triangles, explicitly label which side of one triangle corresponds to which side of the other — a proportion built from mismatched sides gives a confidently wrong answer, not an obviously wrong one.

Common Mistakes

Setting up a proportion with corresponding sides in a mismatched order, such as writing (tree's shadow)/(tree's height) on one side but (person's height)/(person's shadow) on the other.
Keep the same type of measurement in the same position on both sides of the proportion — heights over shadows on one side must match heights over shadows on the other, consistently.
Assuming two triangles are similar just because they look roughly similar in a diagram, without actually verifying a similarity criterion.
Confirm similarity explicitly — usually through AA, since a shared angle plus a parallel-line angle relationship is common in these setups — before trusting any proportion built from the triangles.

Key Takeaways

Similar triangles let an unreachable measurement be found through a proportion built from measurements taken safely and directly.
Indirect measurement using shadows works because the sun's angle creates two similar right triangles at the same moment.
A proportion between similar triangles must consistently match corresponding sides in the same order on both sides.

Summary

Similar triangles turn geometric reasoning into a genuinely practical measurement tool. The next unit returns to right triangles specifically, connecting their side lengths through the Pythagorean Theorem and its converse.

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