What Is a 45-45-90 Triangle?
A 45-45-90 triangle is a right triangle in which both acute angles measure 45°. Because the two non-right angles are equal, the two legs are also equal — making it an isosceles right triangle. It is one of the two special right triangles used most often in geometry and trigonometry, the other being the 30-60-90 triangle.
The fixed angles mean the sides always follow the same ratio regardless of size:
a : a : a√2 (leg : leg : hypotenuse)
Formulas
For a 45-45-90 triangle with leg length a:
| Measurement | Formula | Example (a = 5 cm) |
|---|---|---|
| Leg b | b = a | 5 cm |
| Hypotenuse (c) | c = a√2 | ≈ 7.071 cm |
| Area (A) | A = a² / 2 | 12.5 cm² |
| Perimeter (P) | P = a(2 + √2) | ≈ 17.071 cm |
Reverse formulas
If you know a value other than the leg:
- From hypotenuse: a = c / √2
- From area: a = √(2A)
- From perimeter: a = P / (2 + √2)
How to Use This Calculator
Enter any one known value — leg, hypotenuse, area, or perimeter — and the other three fill in instantly. The formula box updates to show the active relationship. Use the unit selector to switch between metric (mm, cm, m, km) and imperial (in, ft, yd, mi).
Why 45-45-90 Triangles Are Special
Diagonal of a square
Cutting a square along its diagonal creates two identical 45-45-90 triangles. The diagonal length equals the side multiplied by √2 — this is exactly the leg-to-hypotenuse formula.
Trigonometric values at 45°
At 45°, sine and cosine are equal and tangent is 1:
- sin 45° = cos 45° = √2 / 2 ≈ 0.7071
- tan 45° = 1
These values appear throughout mathematics and make 45° calculations particularly clean.
Construction and design
45-45-90 triangles appear in roof framing at a 45° pitch, mitre cuts in carpentry, diagonal bracing in engineering, and tile layouts. Knowing the 1 : 1 : √2 ratio makes it straightforward to find any missing measurement from a single known dimension.
Worked Examples
Example 1 — From the leg
A square tile is 30 cm per side. What is the length of its diagonal?
The diagonal creates a 45-45-90 triangle with leg = 30 cm.
c = 30 × √2 = 30 × 1.41421 ≈ 42.426 cm
Example 2 — From the hypotenuse
A roof rafter runs at 45° and measures 4 m. What is the horizontal run and the vertical rise?
Both are equal to the leg: a = 4 / √2 ≈ 2.828 m
Example 3 — From the area
A right isosceles triangle has an area of 50 cm². Find its dimensions.
- Leg: a = √(2 × 50) = √100 = 10 cm
- Hypotenuse: c = 10√2 ≈ 14.142 cm
- Perimeter: P = 10 × (2 + √2) ≈ 34.142 cm