Special right triangles

Math

Definition

Right triangles with angle measures of 30-60-90 or 45-45-90 whose side ratios are fixed. These ratios allow you to find side lengths without a calculator. In a 45-45-90 triangle the sides are in ratio 1:1:√2, and in a 30-60-90 triangle the sides are 1:√3:2.

Examples

  • A 45-45-90 triangle with legs of 5 has hypotenuse 5√2
  • A 30-60-90 triangle with short leg 4 has long leg 4√3 and hypotenuse 8
  • The diagonal of a square creates two 45-45-90 triangles
Key Fact

45-45-90 → 1:1:√2; 30-60-90 → 1:√3:2

Related Terms

Pythagorean theorem Solving right triangles Sine cosine tangent Unit circle

Study This Concept

Practice special right triangles with free review games in these units:

Geometry Unit 8: Right Triangles and Trigonometry