45 45 90 Triangle Calculator
In a 45 45 90 triangle with legs of 5, the hypotenuse is 5 x sqrt(2), about 7.071. This 45 45 90 triangle calculator solves the entire triangle from a single measured 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 predictable no matter how large or small it is. Measure one side and every other measurement, including area and perimeter, follows automatically. This triangle shows up constantly outside the classroom because it is exactly half of a square, split along its diagonal. That makes it useful for finding a square tile's diagonal, checking a mitered picture-frame corner, working out a roof valley cut at 45 degrees, sizing a staircase stringer, or solving any geometry problem that states one angle is 45 degrees inside a right triangle. Enter whichever side you already know and the calculator returns the rest instantly.
Quick answer
A 45 45 90 triangle is exactly half a square, cut along its 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 exactly half a square, cut along its diagonal.
- •The two legs are always equal, and the hypotenuse is leg x sqrt(2), about 1.41421 times a leg.
- •From the hypotenuse, each leg is hypotenuse divided by sqrt(2), which is the same as hypotenuse x sqrt(2) / 2.
- •The side ratio is always 1 : 1 : sqrt(2), whatever the actual size of the triangle.
- •Because both legs are equal, they double as the triangle's base and height for the area formula.
How to Use
- 1Choose which side you know: a leg (either of the two equal sides) or the hypotenuse (the longest side, opposite the right angle).
- 2Enter its length. Any positive number works, including decimals.
- 3Click Calculate to read the other sides, the area, and the perimeter.
- 4Any unit works, since the calculator only applies the fixed 1 : 1 : sqrt(2) ratio. The outputs use the same unit you entered.
- 5Use the leg value directly if you are solving a diagonal problem, since a square's diagonal is exactly its side length times sqrt(2).
How It Works
Formula
hypotenuse = leg x sqrt(2), area = leg^2 / 2, perimeter = 2 x leg + hypotenuseThe Pythagorean theorem on two equal legs gives hypotenuse squared equals leg squared plus leg squared, or 2 times leg squared, so the hypotenuse is leg times the square root of 2. With legs of 5, that is 5 x 1.41421, or about 7.071. The area is half of leg squared, since the two equal legs serve as the base and height of the right triangle, giving 12.5 for legs of 5. The perimeter adds both legs and the hypotenuse together, which comes out to about 17.071 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) gives the hypotenuse, 25 / 2 gives the area, and adding both legs plus the hypotenuse gives the perimeter.
Hypotenuse of 10
Each leg is 10 divided by sqrt(2), which simplifies to 5 x sqrt(2), or about 7.071. Squaring that leg and halving it gives an area of exactly 25.
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. This is the calculation a tile installer would use to check whether a 12-inch tile fits diagonally into a space.
A small leg of 1
This is the base case for the 1 : 1 : sqrt(2) ratio. Every other 45 45 90 triangle is a scaled version of this one.
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 |
Why this triangle is called special
A 45 45 90 triangle is one of only two special right triangles with a fixed, memorizable side ratio, the other being the 30 60 90 triangle. Because both non-right angles are equal at 45 degrees, the two legs opposite them must also be equal, which is what locks the ratio to 1 : 1 : sqrt(2) regardless of scale.
This predictability is why the triangle appears so often in design and construction. Any time a right angle is split exactly in half, such as a square's diagonal or a symmetric roof peak, the 45 45 90 relationship applies automatically without needing trigonometric tables or a calculator's sine and cosine functions.
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 completely different special triangle.
- Using leg x hypotenuse / 2 for the area. The two legs are the base and height, so the area is leg squared over 2, not leg times hypotenuse over 2.
- Forgetting that the perimeter needs both legs counted separately, not just one leg doubled with the hypotenuse skipped.
- Assuming any right triangle with a diagonal is automatically 45 45 90. The 1 : 1 : sqrt(2) ratio only applies when both acute angles are exactly 45 degrees.
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