1 : 1 : √2

45-45-90 Triangle Calculator

An isosceles right triangle calculator: both acute angles are pinned at 45°, so enter just one side and get the rest from the exact 1 : 1 : √2 ratio.

Isosceles right triangle calculator

Angle α is fixed at 45° (so β is always 45° too, and the two legs are always equal). Enter any one remaining value — a leg, the hypotenuse, the altitude, the area, or the perimeter — pre-loaded here with a leg of 7 mm.

Enter any two known values

Fill in exactly two fields below (a pinned value counts as one). Everything else is solved automatically.

Angle α = 45° (fixed)
Angle mode

Results

45-45-90 isosceles right triangle
Leg a
7 mm
Leg b
7 mm
Hypotenuse c
9.8995 mm
Angle α
45°
Angle β
45°
Altitude h
4.9497 mm
Area
24.5 mm²
Perimeter
23.8995 mm
Inradius
2.0503 mm
Circumradius
4.9497 mm
Isosceles right triangle calculator diagram: 45-45-90 triangle A 45-45-90 isosceles right triangle with both legs 7 millimeters and hypotenuse 9.9 millimeters. A B C a = 6.999999999999999 mm b = 7 mm c = 9.9 mm α = 45° β = 45°

Step-by-step solution

  1. Find the hypotenuse
    c = a / sin(α)
    c = 7 mm / sin(45°)
    = 9.8995 mm
  2. Find the acute angles
    α = atan(a / b), β = 90° − α
    atan(7 / 7)
    = α = 45°, β = 45°
  3. Compute the area
    Area = (1/2)·a·b
    Area = (1/2)·7·7
    = 24.5 mm²
  4. Compute the perimeter
    P = a + b + c
    P = 7 + 7 + 9.8995
    = 23.8995 mm
  5. Compute the altitude to the hypotenuse
    h = (a·b) / c
    h = (7·7) / 9.8995
    = 4.9497 mm
  6. Inradius (radius of the inscribed circle)
    r = (a + b − c) / 2
    r = (7 + 7 − 9.8995) / 2
    = 2.0503 mm
  7. Circumradius (radius of the circumscribed circle)
    R = c / 2
    R = 9.8995 / 2
    = 4.9497 mm

What makes 45-45-90 a "special" right triangle

A 45-45-90 triangle — also called an isosceles right triangle — is a right triangle whose two acute angles are both exactly 45°. Because the angles opposite the two legs are equal, the legs themselves must be equal in length, and the hypotenuse is always the leg length multiplied by √2 (approximately 1.41421356). That single constant is all you need to convert between a leg and the hypotenuse in either direction, with no trigonometry table required.

This calculator pins α (and therefore β) to 45° and lets you supply any one of the six remaining quantities — either leg, the hypotenuse, the altitude to the hypotenuse, the area, or the perimeter — then runs the same general-purpose solver used across this site to fill in everything else.

Worked example: leg = 7

With one leg fixed at 7 mm and both acute angles at 45°, the other leg is also exactly 7 mm, and the hypotenuse is c = 7 × √2 ≈ 9.899 mm. The area is (1/2) × 7 × 7 = 24.5 mm², the perimeter is 7 + 7 + 9.899 ≈ 23.899 mm, the altitude to the hypotenuse is (7 × 7) / 9.899 ≈ 4.950 mm (exactly half the hypotenuse, as it always is in a 45-45-90 triangle), the inradius is (7 + 7 − 9.899)/2 ≈ 2.051 mm, and the circumradius is half the hypotenuse, ≈ 4.950 mm.

Notice that here the altitude to the hypotenuse and the circumradius come out equal — that's not a coincidence specific to this example, but a general property of every 45-45-90 triangle, since both equal exactly half the hypotenuse when the legs are equal.

When you'd use a 45-45-90 triangle

45-45-90 triangles appear whenever a square or a right-angle corner is cut along its diagonal — mitered picture frames, framing carpentry, screen and monitor diagonal calculations, and countless geometry and trigonometry problems. If your triangle instead has a 30° or 60° angle, use the 30-60-90 triangle calculator, or switch between both types in the special right triangle calculator. For a general right triangle with any angle at all, use the full right triangle calculator.

45-45-90 triangle FAQ

What is the exact side ratio of a 45-45-90 triangle?

The two legs are always equal, and the hypotenuse is always the leg length times √2 — a ratio of 1 : 1 : √2. This follows directly from the Pythagorean theorem: if both legs equal x, then c = √(x² + x²) = √(2x²) = x√2.

Why is a 45-45-90 triangle also called an isosceles right triangle?

Isosceles means two sides are equal, and in a 45-45-90 triangle the two legs (opposite the two equal 45° angles) are always the same length — satisfying both the "isosceles" and "right triangle" definitions at once. It is the only right triangle shape that is also isosceles.

How does a 45-45-90 triangle relate to a square?

Drawing one diagonal of a square splits it into two congruent 45-45-90 triangles: the square’s two sides become the equal legs, the diagonal becomes the hypotenuse, and each of the square’s 90° corners is bisected into two 45° angles. This is the quickest way to see why the diagonal-to-side ratio of a square is always √2.

See the full right triangle FAQ for more general questions.

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