Special right triangles have side ratios you can remember, so many problems can be solved without a calculator. Builders use the 3-4-5 triangle to set out square corners; the 1:2 and 1:3 triangles appear in slopes and stairs.
How to use the Special Right Triangles Calculator
- Triangle type
- Short leg a
Results update as you type, and everything is calculated in your browser.
How it works
Choose the triangle and enter the short leg a. The other sides follow from the fixed ratio, and the angles from arctan.
Inputs explained
| Field | What to enter |
|---|---|
| Triangle type | Choose one: 45° – 45° – 90°, 30° – 60° – 90°, Legs 1 : 2 (b = 2a), Legs 1 : 3 (b = 3a) … |
| Short leg a | Units: cm, m, mm, in, ft, yd … |
What each result means
- Hypotenuse c = sqrt([Short leg a]^2 + [Long leg b]^2)
- Smallest angle α = deg(atan([Short leg a]÷[Long leg b]))
- Angle β = 90 – [Smallest angle α]
- Area = [Short leg a]×[Long leg b]÷2
- Perimeter = [Short leg a] + [Long leg b] + [Hypotenuse c]
Example values
How the main result changes when Short leg a changes and everything else stays at the example values above:
| Short leg a (cm) | Hypotenuse c (cm) |
|---|---|
| 1.5 | 3 |
| 2.25 | 4.5 |
| 3 | 6 |
| 3.75 | 7.5 |
| 4.5 | 9 |
| 6 | 12 |
Formula
45-45-90: 1 : 1 : √2 · 30-60-90: 1 : √3 : 2 · 3-4-5: 3 : 4 : 5
Legs 1 : 2 → hypotenuse √5 · legs 1 : 3 → hypotenuse √10
Worked example
Inputs — Triangle type: 30° – 60° – 90°; Short leg a: 3 cm.
Results — Long leg b = 5.196152 cm; Hypotenuse c = 6 cm; Smallest angle α = 30 °; Angle β = 60 °; Area = 7.794229 cm²; Perimeter = 14.196152 cm.
Key terms
- Ratio
- A fixed proportion between lengths.
Frequently asked questions
Why is 3-4-5 useful?
It is the smallest right triangle with whole-number sides – perfect for setting out right angles.
What are the two classic special triangles?
45-45-90 and 30-60-90.
Is any multiple of 3-4-5 a right triangle?
Yes, e.g. 6-8-10 or 30-40-50.
What angle does a 1:2 slope make?
arctan(1/2) ≈ 26.57°.