Square Triangle Calculator
Solve a 45-45-90 right isosceles triangle, also called the "square triangle". Half of a square. Sides in ratio 1 : 1 : ā2.
A "square triangle" is informal shorthand for a 45-45-90 right isosceles triangle: the triangle you get when you cut a square diagonally in half. Two equal sides (the legs), one hypotenuse, two 45Ā° angles and one 90Ā° angle.
The Ratio: 1 : 1 : ā2
If each leg has length 1, the hypotenuse is ā2 ā 1.414. If each leg is 5, the hypotenuse is 5ā2 ā 7.07. Scale up by any factor; the ratio stays the same.
Properties
| Leg | Hypotenuse | Area | Perimeter |
|---|---|---|---|
| 1 | 1.414 | 0.5 | 3.414 |
| 3 | 4.243 | 4.5 | 10.243 |
| 5 | 7.071 | 12.5 | 17.071 |
| 10 | 14.142 | 50 | 34.142 |
Where the Square Triangle Appears
- Diagonal of a square room: measure the side, multiply by ā2.
- Cutting a square in half: the cut line is ā2 times the side length.
- 45Ā° staircase or ramp: the slope length is ā2 Ć horizontal run.
- Bishop's diagonal in chess: moves along 45-45-90 paths.
- Standard set square (drafting): one of two common triangle templates (alongside 30-60-90).
The Math (Pythagoras Simplified)
For an isosceles right triangle, both legs equal each other. Let each leg = a. Pythagoras: aĀ² + aĀ² = cĀ². So 2aĀ² = cĀ², which means c = aā2. The hypotenuse is exactly the leg times ā2.
Use the Full Calculator
The Triangle Calculator handles every scenario described on this page. For the deeper math and worked examples, read the companion guide.