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Special Right Triangles Calculator
Your triangle
Enter the length of any one side — the fixed side ratios fill in the rest. Use the same unit for every side.
Example: a 30-60-90 triangle with a short leg of 5 → long leg 8.66, hypotenuse 10.00, area 21.65, perimeter 23.66.
Behind the numbers
How special right triangles work
Most right triangles need trigonometry to solve, but 30-60-90 and 45-45-90 triangles have fixed side ratios. That means one known side is enough: multiply or divide by the ratio to get every other side. The two acute angles are fixed too, so the area is half the product of the two legs and the perimeter is the sum of all three sides.
Worked example, short leg 5: long leg = 5 × √3 = 8.66. Hypotenuse = 2 × 5 = 10.00. Area = ½ × 5 × 8.66 = 21.65. Perimeter = 5 + 8.66 + 10 = 23.66.
Examples
Example special right triangles
| Triangle | Known side | Other sides | Area | Perimeter |
|---|---|---|---|---|
| 30-60-90 | Short leg 5 | Long leg 8.66, hypotenuse 10.00 | 21.65 | 23.66 |
| 30-60-90 | Long leg 10 | Short leg 5.77, hypotenuse 11.55 | 28.87 | 27.32 |
| 30-60-90 | Hypotenuse 12 | Short leg 6.00, long leg 10.39 | 31.18 | 28.39 |
| 45-45-90 | Leg 7 | Other leg 7.00, hypotenuse 9.90 | 24.50 | 23.90 |
| 45-45-90 | Hypotenuse 10 | Legs 7.07 and 7.07 | 25.00 | 24.14 |
Tips before you solve
The hypotenuse is always the longest side — in a 30-60-90 triangle it is exactly twice the short leg.
The short leg sits opposite the 30° angle and the long leg sits opposite the 60° angle. Mixing them up is the most common mistake.
In a 45-45-90 triangle the two legs are equal, so one leg is all you need to solve the whole triangle.
Keep your units consistent: if the known side is in inches, every answer is in inches, and the area is in square inches.
Good to know
Frequently asked questions
What makes a right triangle “special”?
Only two angle combinations give side ratios you can use without trigonometry: 30-60-90 and 45-45-90. Once you know one side of either triangle, the fixed ratios give you the other two sides exactly.
How do I remember the 30-60-90 side ratio?
Short leg : long leg : hypotenuse = 1 : the square root of 3 : 2. The hypotenuse is exactly twice the short leg, and the long leg is the short leg times the square root of 3 (about 1.73).
What is the 45-45-90 side ratio?
Leg : leg : hypotenuse = 1 : 1 : the square root of 2. The two legs are equal, and the hypotenuse is any leg times the square root of 2 (about 1.41).
Can I use these ratios on any right triangle?
No. The ratios work only when the angles are exactly 30-60-90 or 45-45-90. For any other right triangle, use the general right triangle calculator with the Pythagorean theorem and trigonometry.