Special Right Triangles · Grades 12–12

Special Right Triangles Worksheets

The 45-45-90 and 30-60-90 triangles have fixed side ratios, so missing sides fall out without touching the Pythagorean theorem. Worksheets for both.

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What it is

Two right triangles have fixed side ratios. A 45-45-90 triangle is 1 : 1 : √2; a 30-60-90 triangle is 1 : √3 : 2 (short leg : long leg : hypotenuse).

Why it matters

Special right triangles power exact trigonometry values and appear constantly in geometry, so knowing the ratios saves work and builds intuition.

How to do it

  1. Identify the triangle from its angles.
  2. 45-45-90: the hypotenuse is a leg times √2.
  3. 30-60-90: the hypotenuse is twice the short leg, and the long leg is the short leg times √3.

Examples by level

  • BeginnerIn a 45°-45°-90° triangle with legs 2, the hypotenuse =2√2
  • IntermediateIn a 45°-45°-90° triangle with legs 2, the hypotenuse =2√2
  • AdvancedIn a 30°-60°-90° triangle with short leg 4, the long leg =4√3

Examples are generated by the same engine as the worksheets, so they're always mathematically correct. Built to avoid repeats, not reshuffle the same handful of questions.

Common mistakes

  • Mixing up the 45-45-90 ratio (1:1:√2) with the 30-60-90 ratio (1:√3:2).
  • Putting √3 on the hypotenuse instead of the long leg.
  • Forgetting the hypotenuse is twice the short leg in a 30-60-90.

Tips

  • Label the short leg first in a 30-60-90 triangle.
  • Only the long leg (not the hypotenuse) carries the √3.

For parents

45-45-90: the hypotenuse is a leg times √2. 30-60-90: the hypotenuse is twice the short leg, and the long leg is the short leg times √3.

For teachers

Have students label the short leg first in a 30-60-90; every other side follows from it.

Key vocabulary

hypotenuse
the side opposite the right angle
short leg
in a 30-60-90 triangle, the side opposite the 30° angle

Frequently asked questions

What are the side ratios of special right triangles?
45-45-90 is 1 : 1 : √2 (legs : hypotenuse). 30-60-90 is 1 : √3 : 2 (short leg : long leg : hypotenuse).
How do you find the hypotenuse of a 45-45-90 triangle?
Multiply a leg by √2.

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