45 45 90 Triangle Calculator
In a 45 45 90 triangle with legs of 5, the hypotenuse is 5 x sqrt(2), about 7.07. This 45 45 90 triangle calculator solves the whole triangle from a single side, either a leg or the hypotenuse. The two legs are always equal and the hypotenuse is always sqrt(2) times a leg, which is what makes this special right triangle so useful. Measure one side and the rest follows.
Quick answer
A 45 45 90 triangle is half a square, cut along the diagonal.
Solution
legs 5, hypotenuse 7.0711
Leg
5
Hypotenuse
7.071068
Area
12.5
Perimeter
17.071068
What this tells you
- •A 45 45 90 triangle is half a square, cut along the diagonal.
- •The two legs are equal, and the hypotenuse is leg x sqrt(2), about 1.414 times a leg.
- •From the hypotenuse, each leg is hypotenuse / sqrt(2).
- •The side ratio is always 1 : 1 : sqrt(2), whatever the size.
How to Use
- 1Choose which side you know, a leg or the hypotenuse.
- 2Enter its length.
- 3Read the other sides, the area, and the perimeter.
- 4Any unit works, and the outputs use the same unit.
How It Works
Formula
hypotenuse = leg x sqrt(2), area = leg^2 / 2The Pythagorean theorem on two equal legs gives hypotenuse squared = leg squared + leg squared = 2 leg squared, so the hypotenuse is leg times sqrt(2). With legs of 5, that is 5 x 1.41421 = 7.071. The area is half of leg squared, since the triangle is half a square, giving 12.5 for legs of 5.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Legs of 5
5 x sqrt(2) for the hypotenuse, and 25 / 2 for the area.
Hypotenuse of 10
Each leg is 10 / sqrt(2), which simplifies to 5 x sqrt(2).
The diagonal of a 12-inch square tile
A square's diagonal is its side times sqrt(2), the 45 45 90 relationship in disguise.
45 45 90 Triangle Side Pairs
Legs and their matching hypotenuses.
| Leg | Hypotenuse | Area |
|---|---|---|
| 1 | 1.414 | 0.5 |
| 3 | 4.243 | 4.5 |
| 5 | 7.071 | 12.5 |
| 7 | 9.899 | 24.5 |
| 10 | 14.142 | 50 |
| 12 | 16.971 | 72 |
Common mistakes
- Multiplying the hypotenuse by sqrt(2) to get a leg. Going from hypotenuse to leg divides by sqrt(2), it does not multiply.
- Mixing up the ratios with the 30 60 90 triangle. That one is 1 : sqrt(3) : 2, a different special triangle entirely.
- Using leg x hypotenuse / 2 for the area. The two legs are the base and height, so the area is leg squared over 2.