Similar Triangles
Similar triangles have equal angles and proportional sides.
About this topic
Similar triangles have the same shape but not necessarily the same size: corresponding angles are equal and corresponding sides are in the same ratio, called the scale factor. The shortcuts are AA, SAS similarity and SSS similarity.
Areas scale by the square of the scale factor. Similar triangles let us measure what we cannot reach directly, such as the height of a tree from its shadow, and they underlie trigonometry, maps and scale models.
Worked example: a tree casts a shadow 12 m long while a 2 m pole next to it casts a 3 m shadow. The sun makes the same angle for both, so the triangles are similar by AA. The ratio of tree height to pole height equals 12/3 = 4, so the tree is 8 m tall. If the scale factor is 3, perimeters are 3 times larger but areas are 9 times larger.
Map makers, architects and photographers depend on similarity, and it is the foundation for the sine and cosine ratios. A common error is matching the wrong sides when setting up a proportion; always pair sides opposite equal angles, and write the similarity statement with vertices in order. Another is thinking area scales by the same factor as length. Check your answer by asking whether the larger triangle's side is bigger.
Key ideas
- AA is enough to prove similarity
- Scale factor links corresponding sides
- Areas scale by the square of the factor
Quick check
Use cases: where Similar Triangles helps
- Homework: review the key idea behind Similar Triangles before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Similar Triangles without notes.
- Visual learning: play with the interactive lab to see how Similar Triangles changes when you adjust its inputs.
- Teaching a friend: explain Similar Triangles in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Similar Triangles?
Similar triangles have equal angles and proportional sides.
What will I learn in Similar Triangles?
AA is enough to prove similarity. Scale factor links corresponding sides. Areas scale by the square of the factor.
Is Similar Triangles part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Similar Triangles lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Similar Triangles?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.