Math
45-45-90 Triangle Calculator
A 45-45-90 triangle is an isosceles right triangle: both legs are equal and the hypotenuse is √2 times a leg.
Enter either a leg or the hypotenuse to get the other side as an exact radical and decimal, plus area, perimeter, and the height to the hypotenuse.
Any positive length.
Try an example
Result
Hypotenuse
—
- Each leg
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- Area
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- Perimeter
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- Height to hypotenuse
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Study path
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What a 45-45-90 triangle is
Cut a square along a diagonal and you get two 45-45-90 triangles. The two legs are the square's sides and the The longest side of a right triangle, always opposite the right angle. is the diagonal, which is why the hypotenuse is always √2 times a leg. Every 45-45-90 triangle is similar to every other, so one side fixes all three.
| Known | Leg | Hypotenuse | Area |
|---|---|---|---|
| Leg = 1 | 1 | √2 ≈ 1.4142 | 0.5 |
| Leg = 5 | 5 | 5√2 ≈ 7.0711 | 12.5 |
| Hypotenuse = 10 | 5√2 ≈ 7.0711 | 10 | 25 |
| Hypotenuse = 4 | 2√2 ≈ 2.8284 | 4 | 4 |
How to use it
- Choose whether you know a leg or the hypotenuse.
- Enter the length.
- Read the missing side in The answer kept as a fraction or radical instead of a rounded decimal. (with √2) and as a decimal, along with the area and perimeter.
- Copy the ratio steps if you need to show work.
How to read the answer
When the leg is a whole number the hypotenuse is that number times √2 — A number that cannot be written as a fraction of two whole numbers, like π or √2., so the decimal is rounded. When the hypotenuse is a whole number the legs are that number times √2/2. The area from a hypotenuse h is simply h²/4.
Common mistakes and edge cases
- Forgetting to rationalize: h/√2 should be written h√2/2.
- Using √3 instead of √2 — that ratio belongs to the 30-60-90 triangle.
- Assuming the hypotenuse is twice the leg. It is about 1.414 times the leg.
- A zero or negative side is not a triangle.
Worked examples
Leg 7
Hypotenuse 7√2
Hypotenuse
7√2 (9.8995)
Leg 1
Unit triangle, hypotenuse √2
Hypotenuse
√2 (1.4142)
Hypotenuse 10
Legs 5√2
Hypotenuse
10
Decimal leg 2.5
Hypotenuse 5√2/2
Hypotenuse
5√2/2 (3.5355)
Negative side
Hypotenuse
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Frequently asked questions
What are the side ratios of a 45-45-90 triangle?+
1 : 1 : √2. Both legs are equal and the hypotenuse is a leg times √2.
How do I find a leg from the hypotenuse?+
Divide by √2 and rationalize: leg = h/√2 = h√2/2. For hypotenuse 8 the legs are 4√2 ≈ 5.657.
Is a 45-45-90 triangle isosceles?+
Yes. It is the only right triangle that is also isosceles, because the two acute angles are both 45°.
What is the area of a 45-45-90 triangle from the hypotenuse?+
Area = h²/4. With h = 10 the area is 25. From a leg x the area is x²/2.
Why is the hypotenuse x√2?+
Pythagorean theorem: x² + x² = 2x², and √(2x²) = x√2.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026