SOLVE RIGHT TRIANGLE GUIDE
30–60–90 and 45–45–90 triangles
Some right triangles have fixed angles that produce useful exact side ratios. These shortcuts can simplify an answer and provide an independent check on a calculator.
The 45–45–90 triangle
The acute angles are equal, so the legs are equal. If each leg has length x, Pythagoras gives c² = x² + x² = 2x².
Hypotenuse = leg × √2
Example: two legs of 7 cm
The hypotenuse is 7√2 cm, approximately 9.8995 cm. The area is 7 × 7 / 2 = 24.5 cm².
If the hypotenuse is known instead, divide it by √2 to find either leg.
The 30–60–90 triangle
The shortest side faces the 30° angle. The longer leg faces 60°, and the hypotenuse faces 90°.
Short leg = hypotenuse / 2
Long leg = short leg × √3
Example: hypotenuse of 10 m
The short leg is 5 m. The long leg is 5√3 m, approximately 8.6603 m. The area is 25√3 / 2 m², approximately 21.6506 m².
Exact and decimal answers describe the same length
An expression such as 5√3 preserves an exact value. A decimal such as 8.66 is rounded. Keep the exact expression or full calculator precision when the value feeds another calculation.
These are different from Pythagorean triples
A 3–4–5 triangle is a Pythagorean triple because all three sides are whole numbers and satisfy a² + b² = c². Its acute angles are approximately 36.87° and 53.13°, so it is neither of these special-angle triangles.
Try a 45–45–90 triangleFormula reference
For an independent explanation of these relationships, see Pearson’s right triangle formulas and worked examples.
Related right triangle guides
- How to find the hypotenuse
- How to find a missing right triangle angle
- SOH CAH TOA, explained visually
- Right triangles in practical measurements
- How to find a missing side of a right triangle
Open the right triangle calculator to check your own measurements.