A 45 45 90 triangle always has a right angle (90°) and two equal angles (45° and 45°). If you know one leg length, the calculator below computes the other leg and the hypotenuse using fixed ratios.
This article explains the exact formulas, what each side means, and how to use the results for real measurements like stair runs, diagonal braces, and ramps.
What Is a 45 45 90 Triangle?
A 45 45 90 triangle is a special right triangle where the angles are 45°, 45°, and 90°. Because two angles are equal, the two legs (the sides touching the right angle) are also equal in length.
That fixed geometry makes the math simple: the side lengths always follow the same proportions.
45 45 90 Triangle Ratios (The Key Facts)
Let the legs be a and b, and the hypotenuse be c. In a 45 45 90 triangle:
- a = b (the two legs are equal)
- c = a√2 (the hypotenuse is the leg times √2)
- Angles are fixed: 45°, 45°, 90°
Because the ratios never change, one input is enough to compute the rest.
Variables Used in This Calculator
To avoid confusion, here is how the calculator labels the sides:
| Triangle Part | Meaning | Symbol |
|---|---|---|
| Leg (given) | One of the two equal sides | a |
| Leg (other) | The other equal side | b |
| Hypotenuse | The side across from 90° | c |
Formulas the Calculator Uses
Choose a known leg length a. Then the triangle’s other side lengths are:
- b = a
- c = a × √2
Angles are constant, so the calculator outputs:
- 90° (right angle)
- 45° and 45° (the equal acute angles)
Unit Handling: Keeping Measurements Consistent
The calculator supports common length units. Internally, it converts your input to a base unit, applies the √2 ratio, and converts results back to your selected output unit.
That means you can enter inches and receive centimeters, or enter meters and receive millimeters—without doing the conversion math yourself.
How to Use the 45 45 90 Triangle Calculator
- Enter the known leg length (one of the equal sides).
- Select the unit for that input (for example, inches or meters).
- Choose the output unit you want for the results.
- Click Calculate to get the other leg and the hypotenuse.
If you leave a required field blank or enter a non-positive number, the calculator shows an error so you can correct it.
Practical Examples (Real-World Use Cases)
Example 1: Diagonal Bracing
Suppose you need a brace that forms a 45 45 90 triangle. If one leg (the distance along one beam) is 12 in, then the other leg is also 12 in. The hypotenuse (the diagonal brace length) is 12√2 ≈ 16.97 in.
This is why diagonal supports often come out slightly longer than the side distance.
Example 2: Stair or Ramp Planning
If a stair run and rise form a 45 45 90 relationship, the rise equals the run. For a 30 cm run, the rise is 30 cm and the diagonal distance (the stringer length) is 30√2 ≈ 42.43 cm.
Using the calculator helps you size the diagonal member correctly.
Common Mistakes to Avoid
- Mixing up hypotenuse and legs: The hypotenuse is opposite the 90° angle.
- Forgetting the legs are equal: In a 45 45 90 triangle, both legs match.
- Using the wrong unit: Always check the input unit and desired output unit.
Frequently Asked Questions
What is the hypotenuse formula for a 45 45 90 triangle?
In a 45 45 90 triangle, the hypotenuse equals the leg multiplied by √2. If one leg is a, then the hypotenuse is c = a√2. The two legs are equal, so the other leg is b = a.
If I know the hypotenuse, how do I find a leg?
If you know the hypotenuse c, then each leg is c/√2. Since √2 is about 1.4142, you can divide by 1.4142 to get the leg length. The calculator focuses on entering a leg, but the relationship is fixed.
Are the angles always exactly 45°, 45°, and 90°?
Yes. A 45 45 90 triangle is defined by those three angles. As long as the triangle is truly a 45-45-90 right triangle, the angles are fixed. Only the side lengths scale up or down together.
Can this calculator convert between inches, centimeters, and meters?
Yes. You choose the input unit and the output unit. The calculator converts the given leg to a base unit, applies the √2 ratio, and then converts the results into your chosen output unit. This prevents manual conversion errors.
What should I do if my input is zero or negative?
Triangle side lengths must be positive. If you enter zero or a negative number, the calculator flags it as invalid and asks for a positive value. Re-enter a real length measurement, such as 2, 10, or 0.5.
Bottom Line
A 45 45 90 triangle is predictable: two equal legs and a hypotenuse that is each leg times √2. Enter one leg length into the calculator to get the other leg and the hypotenuse immediately, with correct unit conversion.
