Isosceles Right Triangle Calculator - 45-45-90 Side, Area, Perimeter

An isosceles right triangle calculator solves the special 45-45-90 right triangle, where the two legs that meet at the right angle are equal. Type one known measurement (a leg or the hypotenuse) and the page returns the equal legs, the hypotenuse, the area, the perimeter, the altitude to the hypotenuse, and the inradius and circumradius in one pass.

• • Geometry homework and special right triangles: Confirm 45-45-90 problems that ask for the hypotenuse, area, or perimeter of an isosceles right triangle, including scaled Pythagorean triples like 1-1-sqrt(2) and 5-5-5*sqrt(2).
• • Carpentry and construction with square corners: Find the diagonal of a square panel, the rafter length of a 45 degree roof pitch, or the miter length for a 45 degree joint.
• • Tiling and craft layout: Work out the cut length of a diagonal strip across a square tile, the corner-to-corner distance on a rectangular pattern, or the boundary length of a half-square quilt block.
• • Engineering, surveying, and physics preliminaries: Compute the magnitude of an equal-component 2D vector, the diagonal of a square sensor, or the resultant of two perpendicular forces of equal size.

The page is intentionally narrow: it covers the 45-45-90 case. For a general right triangle whose legs are not equal, the dedicated right-triangle-calculator is the next click. For an isosceles triangle that does not have a right angle, the isosceles-triangle-area-calculator covers that case.

For a general right triangle whose legs are not equal, the Right Triangle Calculator solves the right-triangle case from any two values.

The page implements the 45-45-90 formulas directly. Given a leg a, the hypotenuse is a times the square root of 2; given a hypotenuse h, the leg is h divided by the square root of 2.

• a: Length of one of the two equal legs (the sides that meet at the right angle).
• h = a * sqrt(2): Length of the hypotenuse, the longest side, opposite the 90 degree angle.
• A = a^2 / 2: Area of the triangle, half the product of the two legs.
• P = a * (2 + sqrt(2)): Perimeter, the sum of the two equal legs and the hypotenuse.
• m = h / 2: Altitude from the right angle to the hypotenuse; equals h / 2 for any right triangle.
• r = a * (2 - sqrt(2)) / 2: Inradius, the radius of the inscribed circle.
• R = h / 2: Circumradius, the radius of the circumscribed circle.

The square root of 2 is irrational, so the hypotenuse and perimeter only land on clean decimals when the leg or hypotenuse is a multiple of sqrt(2). The page shows four decimal places, which is enough precision for carpentry, tiling, and homework.

According to Wolfram MathWorld, an isosceles right triangle has legs of length a, hypotenuse a * sqrt(2), area a^2 / 2, and perimeter a * (2 + sqrt(2)).

According to Wolfram MathWorld, the Pythagorean theorem states that for any right triangle, a^2 + b^2 = c^2; setting a = b gives c = a * sqrt(2), which is the 45-45-90 relationship.

The 5-5-5*sqrt(2) example is a scaled copy of the (1, 1, sqrt(2)) family, and the Pythagorean Triples Calculator page is the right next step if you want to see the integer side family (3-4-5, 5-12-13, and so on) that the Pythagorean theorem also produces.

Four ideas explain why the formulas look the way they do.

An isosceles right triangle is the same shape as a square cut along its diagonal: the square's two sides are the equal legs, and the cut is the hypotenuse of length side * sqrt(2).

If you want to see the integer side family (3-4-5 and similar), the Pythagorean triples page covers the right triangle family without the 45-45-90 constraint.

An isosceles right triangle is the special case where the two equal sides also meet at 90, and the Isosceles Triangle Area Calculator page covers the isosceles case without the right-angle constraint.

Four short steps cover the most common cases.

1. 1 Pick the value you already know: Select 'Leg length (a)' or 'Hypotenuse (h)'. The default is leg.
2. 2 Type the value: Enter the known leg or hypotenuse. Values must be positive.
3. 3 Choose a unit: Pick cm, m, inches, or feet. The area shows in the matching unit squared.
4. 4 Read every derived value: The results panel shows the equal legs, hypotenuse, area, perimeter, altitude to the hypotenuse, and the inradius and circumradius.

Try the leg a = 12 in (a one-foot square). The page shows both legs 12 in, hypotenuse 16.9705627485 in, area 72 in^2, perimeter 40.9705627485 in, altitude to the hypotenuse 8.4852813742 in, inradius 3.5147186258 in, and circumradius 8.4852813742 in. The hypotenuse is the diagonal of a 12 inch square and the circumradius is half that diagonal, as expected.

If you only need the area in the chosen unit squared, the Area Calculator page handles area for many other shapes in the same category.

These benefits matter when the math has to be right the first time and you only have one measurement in hand.

• • Solve the whole triangle from one number: A single leg or hypotenuse is enough. The page returns the other side, the area, the perimeter, the altitude to the hypotenuse, and the inradius and circumradius without a second pass.
• • Avoid arithmetic mistakes with sqrt(2): Computing a * sqrt(2) by hand is where most 45-45-90 mistakes happen. The page keeps the irrational part exact and only rounds for display.
• • Works in metric and imperial: Centimeters, meters, inches, and feet are all first-class. The area automatically shows in cm^2, m^2, in^2, or ft^2.
• • Confirms the standard 45-45-90 ratios: If you only remember that a 45-45-90 triangle has legs in the ratio 1:1:sqrt(2), the page lets you plug in one number and read back the rest.
• • Great for square diagonals and 45 degree cuts: The hypotenuse output is the diagonal of a square with the given side, the altitude to the hypotenuse is half the diagonal, and the circumradius is the radius of the smallest circle that contains the square.
• • Connects to the rest of geometry: If your problem is a right triangle that is not 45-45-90, the right-triangle-calculator handles it. If it is an isosceles triangle without a right angle, the isosceles-triangle-area-calculator covers that case.

Use it to confirm the diagonal of a square, the rafter length for a 45 degree roof pitch, or a homework answer that came out close to but not exactly a * sqrt(2).

The 45-45-90 triangle is the right-triangle special case of the equal-side family, and the Equilateral Triangle Area Calculator page is the other classical equal-side case.

A few factors change how the result should be read.

• • This page is only for the 45-45-90 case. For a right triangle whose legs are not equal, the right-triangle-calculator solves the general case from any two values.
• • The page assumes a flat Euclidean triangle. It does not compute spherical geometry, the great-circle distance, or the path length along a curve.
• • The result is the ideal geometric value. Real-world cut lengths for carpentry, tile, or fabric need to add kerf (saw blade width) or seam allowance.

If the page rejects an input that you know is correct, double-check that the value is positive, that you picked the right input type (leg vs. hypotenuse), and that the unit is the one you intended.

According to Wolfram MathWorld, the area of a right triangle is half the product of the two legs, and the circumradius of any right triangle is half the hypotenuse.

For a triangle whose sides and angles are not constrained to the 45-45-90 case, the Triangle Calculator page handles the general triangle with all three sides, all three angles, the area, and the perimeter.