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.
Generate special right triangles worksheets →Free plan · 2 credits every month · answer keys included · no card required.
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
Once you recognize a 45-45-90 or 30-60-90 triangle, the side lengths fall out of fixed ratios instead of a Pythagorean calculation, a shortcut that pays off constantly once trigonometry starts asking for exact values instead of decimals.
How to do it
- Identify the triangle from its angles.
- 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.
Examples by level
- BeginnerIn a 45°-45°-90° triangle with legs 2, the hypotenuse =Answer2√2
- IntermediateIn a 45°-45°-90° triangle with legs 2, the hypotenuse =Answer2√2
- AdvancedIn a 30°-60°-90° triangle with short leg 4, the long leg =Answer4√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.
Ready to practice?
Generate a print-ready special right triangles pack with answer keys: pick a grade and difficulty, and it's ready in seconds.
Create free worksheetsFree to start · answer keys included · free accounts get 14 days of Standard · no card required.