45-45-90 Right Triangle Calculator

A 45-45-90 triangle is a special right triangle with interior angles of exactly 45°, 45°, and 90°. Because two of its angles are equal, it's also an isosceles triangle, which means its two legs are the same length.

It shows up constantly in geometry, construction, and design. Any time you cut a square diagonally from corner to corner, you get two 45-45-90 triangles. That's actually the easiest way to picture it.

What makes it "special" isn't just the angles. It's the fact that the side lengths always follow a predictable ratio, so you only need one measurement to find all three sides.

Solving a 45-45-90 triangle means finding all three side lengths when you're given at least one of them. Because the angles are fixed, the ratios between the sides never change, so the process is straightforward.

There are really only two scenarios you'll run into:

• You know one of the legs and want to find the hypotenuse (and the other leg, which is the same).
• You know the hypotenuse and want to find each leg.

In both cases, you're applying the same core ratio. No need for a full trigonometric calculation. The fixed 1 : 1 : √2 relationship handles everything, and the formulas below make it mechanical.

The sides of a 45-45-90 triangle always relate to each other in the ratio 1 : 1 : √2. The two legs are equal, and the hypotenuse is exactly √2 times the length of either leg.

This ratio holds true no matter how big or small the triangle is. A tiny triangle on graph paper and a massive triangular roof truss follow the exact same proportions.

All the formulas for a 45-45-90 triangle flow from that 1 : 1 : √2 ratio. Let a represent the length of each leg and c represent the hypotenuse. The two core relationships are:

• Hypotenuse: c = a√2
• Leg: a = c / √2 = c√2 / 2

Everything else, including area, perimeter, and radius values, builds on these two expressions.

To use a 45-45-90 triangle calculator, you input one known side and the tool returns all the others. Here's how the math works step by step for each starting point:

1. Given a leg (a): The other leg equals a. The hypotenuse equals a × 1.4142. Area = a² / 2. Perimeter = a(2 + √2).
2. Given the hypotenuse (c): Each leg equals c × 0.7071. Area = c² / 4. Perimeter = c(1 + √2).

Walk through a quick example. Suppose a leg is 5 inches long:

• Other leg: 5 inches (same, because 45-45-90)
• Hypotenuse: 5 × √2 ≈ 7.071 inches
• Area: (5 × 5) / 2 = 12.5 square inches
• Perimeter: 5 + 5 + 7.071 ≈ 17.071 inches

Every calculator following this triangle type uses exactly those steps under the hood.

Because the two legs are equal and they meet at the right angle, calculating the area is unusually clean. The legs themselves serve as the base and height, so you don't need to do any extra work to find an altitude.

The perimeter is just the sum of all three sides. With two equal legs of length a and a hypotenuse of a√2, you're adding a + a + a√2. That factors neatly.

Beyond the basic sides, there are a few other measurements that come up in geometry problems and real-world applications.

Height (altitude to the hypotenuse): For a 45-45-90 triangle with legs of length a, the altitude drawn from the right angle to the hypotenuse has a length of a / √2 = a√2 / 2. This is the same as the leg of the triangle divided by √2, or equivalently half the hypotenuse.

Inradius (r): The inradius is the radius of the largest circle that fits inside the triangle. For this triangle:

r = a(√2 - 1) / √2, which simplifies to approximately r ≈ 0.2929 × a.

You can also express it as r = (a + a - c) / 2 = a - (a√2 / 2), using the general inradius formula r = (sum of legs - hypotenuse) / 2.

Circumradius (R): The circumradius is the radius of the circle that passes through all three vertices. For any right triangle, the circumradius equals half the hypotenuse:

R = c / 2 = a√2 / 2

So the circumcenter sits right at the midpoint of the hypotenuse, which is a neat geometric property all right triangles share.

A few worked examples to tie everything together:

Notice that when the leg doubles (5 to 10), the area quadruples (12.5 to 50). That's a property of area scaling with the square of linear dimensions, and it applies to all similar figures, not just this triangle type.

These two are the most common special right triangles, and they often get compared in math courses. They're similar in that both have fixed angle sets and predictable side ratios, but they behave differently.

The 45-45-90 is generally easier to work with because of its symmetry. The 30-60-90 has three distinct side lengths, so you have to track which side is the short leg, long leg, or hypotenuse more carefully.

Both are worth memorizing because they appear constantly in standardized tests, geometry proofs, and practical applications like carpentry and engineering.

This triangle shows up in more places than most people realize. A few real and practical examples:

• Square diagonals: Cut any square along its diagonal and you get two 45-45-90 triangles. This is how the triangle is derived in the first place, and it makes it essential for understanding square geometry.
• Coordinate geometry: Lines with slopes of 1 or -1 create 45° angles with the axes. Triangles formed under those lines are 45-45-90 triangles, which makes distance and area calculations straightforward.
• Trigonometry: The exact values sin(45°) = cos(45°) = √2/2 come directly from this triangle. Those values show up repeatedly in calculus, physics, and engineering.
• Construction and carpentry: Cutting 45° mitered corners for picture frames or baseboards relies on this triangle's geometry. A speed square set to 45° is essentially a physical 45-45-90 triangle.
• Roof framing: Hip roofs often use 45° angles at the corners, where the geometry of the 45-45-90 triangle determines rafter lengths.
• Navigation and surveying: Bearing changes of 45° create this triangle when broken down into horizontal and vertical components.

Understanding this triangle isn't just an academic exercise. It's a practical tool that keeps showing up whenever symmetry, right angles, and 45° directions come together.