Special Right Triangles
45-45-90 and 30-60-90 triangles have fixed side ratios.
About this topic
Two right triangles have fixed side ratios. The 45°-45°-90° triangle has legs equal and hypotenuse = leg × √2, with sides in ratio 1 : 1 : √2. The 30°-60°-90° triangle has sides in ratio 1 : √3 : 2, with the shortest leg opposite 30°.
Knowing these lets you find missing sides without a calculator, and gives the exact trig values for 30°, 45° and 60°. They arise from cutting a square along its diagonal or an equilateral triangle in half.
Worked example: a square has a diagonal; if the side is 6, the diagonal is 6√2 ≈ 8.49. In a 30-60-90 triangle with hypotenuse 10, the short leg is 5 and the long leg is 5√3 ≈ 8.66. If the long leg is 9, the short leg is 9/√3 = 3√3 and the hypotenuse is 6√3. Always identify which side is the short leg first, then scale the others from it.
These ratios are used in roof framing, tiling at 45° angles, and in setting up the unit circle in trigonometry. A common error is mixing up where √2 and √3 belong: in the 45 triangle only the hypotenuse gets √2, and in the 30-60-90 triangle the long leg gets √3 while the hypotenuse is simply double the short leg. Draw a quick sketch and label sides 1, √3 and 2 to avoid mistakes. Keep answers in exact form unless asked for a decimal.
Key ideas
- 45-45-90 sides: 1 : 1 : √2
- 30-60-90 sides: 1 : √3 : 2
- Skip the Pythagorean theorem
Quick check
Use cases: where Special Right Triangles helps
- Homework: review the key idea behind Special Right Triangles before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Special Right Triangles without notes.
- Visual learning: play with the interactive lab to see how Special Right Triangles changes when you adjust its inputs.
- Teaching a friend: explain Special Right 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 Special Right Triangles?
45-45-90 and 30-60-90 triangles have fixed side ratios.
What will I learn in Special Right Triangles?
45-45-90 sides: 1 : 1 : √2. 30-60-90 sides: 1 : √3 : 2. Skip the Pythagorean theorem.
Is Special Right 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 Special Right 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 Special Right 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.