45-45-90 Triangle Calculator
This calculator finds every measurement of a 45-45-90 triangle, a right isosceles triangle with two 45 degree angles and one 90 degree angle, from a single known value. Because both legs are always equal and the hypotenuse is always exactly the leg length multiplied by the square root of 2, knowing just one side, or even the area or the perimeter, is enough to work out everything else. Choose what you know from the dropdown, leg length, hypotenuse, area, or perimeter, enter that one figure, and the calculator instantly returns the leg, hypotenuse, area, and perimeter, plus the height to the hypotenuse, inradius, and circumradius in a full measurements table. A worked line underneath shows the exact formula used for your chosen input, so you can see how each figure is derived rather than just reading off a number. All calculations use the full precision of the square root of 2 rather than a rounded decimal, so results stay accurate even when you are chaining several conversions together. This is handy for geometry homework, checking a diagonal cut, laying out a mitred joint, or framing a roof at a 45 degree pitch, where the classic root two ratio turns up again and again. Because the shape has fixed angles, one measurement is always enough, you never need to enter two values to get a full solution.
Known Value
45-45-90 Triangle
Both legs (a) are equal. Hypotenuse c = a√2.
All Measurements
Area and Height
What is a 45-45-90 Triangle?
A 45-45-90 triangle is a special right triangle with two 45-degree angles and one 90-degree angle. Because both acute angles are equal, the triangle is also isosceles, meaning both legs (the two shorter sides) are identical in length. The hypotenuse (the side opposite the right angle) is always exactly the leg length multiplied by the square root of 2.
This triangle appears constantly in geometry, trigonometry, construction, and design. It is the shape you get when you cut a square diagonally in half. The 45-45-90 triangle is one of only two special right triangles with fixed angle ratios (the other being the 30-60-90 triangle).
The 45-45-90 Triangle Formulas
| Measurement | Formula | Example (leg = 5) |
|---|---|---|
| Leg (a) | a (given, or derived below) | 5 |
| Hypotenuse (c) | c = a × √2 | 5 × 1.41421 = 7.07107 |
| Area | Area = a² / 2 | 5² / 2 = 12.5 |
| Perimeter | P = 2a + a√2 = a(2 + √2) | 5 × 3.41421 = 17.07107 |
| Height to hypotenuse (h) | h = a / √2 = c / 2 | 5 / 1.41421 = 3.53553 |
| Inradius (r) | r = a(√2 - 1) / √2 = (a / √2) × (√2 - 1) | 5 × 0.29289 = 1.46447 |
| Circumradius (R) | R = c / 2 = a√2 / 2 | 7.07107 / 2 = 3.53553 |
Deriving the Formulas from Pythagoras
The 45-45-90 ratios follow directly from the Pythagorean theorem. If both legs are equal to a, then:
a² + a² = c²
2a² = c²
c = √(2a²) = a√2
This is why the hypotenuse is always exactly the leg times the square root of 2, regardless of the actual size of the triangle. The ratio is exact, not an approximation. When someone says "multiply by 1.41421", they are using a rounded decimal approximation of √2. The calculator above uses the full precision of Math.SQRT2 in JavaScript, which is accurate to 15 significant figures.
Worked Example Using the Default Values
With a leg length of 5 units:
- Hypotenuse = 5 × √2 = 7.07107 units
- Area = 5² / 2 = 25 / 2 = 12.5 square units
- Perimeter = 5 + 5 + 7.07107 = 17.07107 units
- Height to hypotenuse = 5 / √2 = 3.53553 units
Finding the Leg from the Hypotenuse
If you know the hypotenuse c and need the leg, rearrange c = a√2:
a = c / √2 = c × √2 / 2
For example, if the hypotenuse is 10: a = 10 / √2 = 10 × √2 / 2 = 5√2 = approximately 7.07107 units.
Finding the Leg from the Area
Area = a² / 2, so a² = 2 × Area, giving a = √(2 × Area).
For example, if the area is 50 square units: a = √(2 × 50) = √100 = 10 units.
Real-World Uses of the 45-45-90 Triangle
- Roof framing: a 45-degree pitch means equal rise and run, forming a 45-45-90 triangle.
- Tile cutting: diagonal cuts across square tiles create two 45-45-90 triangles.
- Navigation and surveying: bearings at 45 degrees and equal north/east displacements.
- Computer graphics: pixel-perfect diagonal lines on a grid at 45 degrees.
- Woodworking: mitre joints at 45 degrees join two boards at a right angle.
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Method: All calculations use exact arithmetic with JavaScript's Math.SQRT2 constant (√2 = 1.41421356237...) and Math.sqrt(). No intermediate rounding is applied. Results are displayed rounded to 5 decimal places for display purposes.