Special Right Triangles
45-45-90 and 30-60-90 have fixed side ratios.
Key idea
45-45-90 and 30-60-90 have fixed side ratios.
In depth
Two right triangles have fixed ratios: 45-45-90 has sides 1 : 1 : √2, and 30-60-90 has sides 1 : √3 : 2. They arise from halving a square and an equilateral triangle, and let you find sides without a calculator.
Worked example
A 45-45-90 triangle with legs 5 has hypotenuse 5√2. A 30-60-90 triangle with short side 4 has hypotenuse 8 and long side 4√3.
Formula
Real-world uses
- Calculating the diagonal of a square tile or the height of an equilateral roof
Common mistake
Placing √3 on the wrong leg. It is opposite the 60° angle.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Special Right Triangles?
45-45-90 and 30-60-90 have fixed side ratios. It is a topic at the intermediate level in Geometry.
What is the formula for Special Right Triangles?
The key formula is 1 : 1 : √2 and 1 : √3 : 2.
Where is Special Right Triangles used in real life?
Real-world examples: Calculating the diagonal of a square tile or the height of an equilateral roof.
What should I learn before Special Right Triangles?
Make sure you are comfortable with Pythagorean Triples, the previous topic in Geometry.
What is a common mistake with Special Right Triangles?
Placing √3 on the wrong leg. It is opposite the 60° angle.
How can I practise Special Right Triangles?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Special Right Triangles?
Move on to Perimeter next.
Related intermediate topics in Geometry
- Vertical Angles – Opposite angles formed by intersecting lines are equal.
- Parallel Lines and Transversals – Corresponding and alternate interior angles are equal.
- Exterior Angle Theorem – An exterior angle equals the sum of the two remote interior angles.
- Triangle Inequality – Any two sides sum to more than the third.
- Congruent Triangles – Same shape and size: SSS, SAS, ASA, AAS, HL.
- Similar Triangles – Same shape, proportional sides: AA, SAS~, SSS~.