Isosceles Right Triangle Hypotenuse - 45-45-90 Side Solver

An isosceles right triangle hypotenuse is the longest side of a 45-45-90 right triangle, the side across from the 90 degree corner that stretches between the two 45 degree corners. The two shorter sides, called the legs, are equal in length, so the hypotenuse is forced into a fixed ratio: h = a * sqrt(2).

• • Geometry homework and special right triangle problems: Confirm 45-45-90 problems that ask for the hypotenuse, area, or perimeter, including scaled versions like 5-5-5*sqrt(2).
• • Carpentry and construction with square corners: Find the diagonal of a square panel, a 45 degree rafter length, or a 45 degree miter length.
• • Tiling, crafts, and quilt layout: Work out the cut length of a diagonal strip across a square tile or block.
• • Engineering, surveying, and physics preliminaries: Compute the magnitude of an equal-component 2D vector or the resultant of two perpendicular forces of equal size.

In practice the hypotenuse is the diagonal of a square: for a 5 cm square it is 5 * sqrt(2) = 7.0710678 cm. The page handles both directions, so h = a * sqrt(2) when you know a leg and a = h / sqrt(2) when you know the hypotenuse.

If the problem also asks for the inradius, circumradius, or altitude to the hypotenuse, the Isosceles Right Triangle page extends the same leg input to those extra measurements.

The page applies the Pythagorean theorem to two equal legs, then collapses the formula using the fact that a = b in a 45-45-90 triangle. The hypotenuse is a times sqrt(2), and the same page works in reverse when the hypotenuse is the known side.

• 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 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.

The leg-to-hypotenuse ratio is fixed because the two legs are equal. Squaring both legs and adding gives a^2 + a^2 = 2 * a^2, and the square root of that is a * sqrt(2). When the hypotenuse is the known side, the same relationship solves for the leg as a = h / sqrt(2); the area and perimeter still come from the leg.

According to Wolfram MathWorld, an isosceles right triangle has legs of length a and a hypotenuse of length a*sqrt(2), so the only isosceles right triangle is the 45-45-90 triangle.

When the triangle stops being isosceles and the two legs differ, the Hypotenuse Calculator page applies c = sqrt(a^2 + b^2) to two unequal legs and reports the missing leg in either direction.

Four ideas explain why the hypotenuse is locked to a * sqrt(2), why the ratio never changes, and why the formula works in both directions.

These four ideas explain why the hypotenuse is a*sqrt(2) and why a hypotenuse input recovers the leg with one division.

When the triangle is still a right triangle but the two legs are not equal, the ABC Triangle page uses the same Pythagorean theorem in the classic a, b, c notation to recover whichever value is missing.

Six short steps cover both directions, from a textbook leg input to a square diagonal measured with a tape.

1. 1 Pick the Solve for mode: Choose Hypotenuse when the known side is a leg, and Leg when the only known side is the hypotenuse (a square panel diagonal).
2. 2 Choose a length unit: Pick centimeters, meters, inches, or feet. The hypotenuse, area, and perimeter come back in the matching linear unit or unit squared.
3. 3 Type the side you already know: Enter the leg in hypotenuse mode, or the hypotenuse in leg mode. The default of 5 in hypotenuse mode gives the 5-5-5*sqrt(2) textbook triangle.
4. 4 Read the hypotenuse first: The result panel shows the hypotenuse, the leg, the area, the perimeter, and the side ratio 1 : 1 : sqrt(2).
5. 5 Switch the mode to check the reverse: To confirm a leg-to-hypotenuse result, switch Solve for to Leg and re-enter the hypotenuse; the page gives back the original leg.
6. 6 Reset for the next problem: Click Reset to return to the 5 cm hypotenuse-mode default. The page keeps the same unit across resets.

For a 5 cm square tile, leave Solve for on Hypotenuse and type 5. The page returns hypotenuse 7.0710678 cm, leg 5 cm, area 12.5 cm^2, perimeter 17.0710678 cm.

When the triangle stops being a right triangle and the equal sides are not perpendicular, the Isosceles Triangle Side Calculator page solves the same two-equal-sides setup without the 90 degree corner.

These benefits matter most when a hand calculation is the alternative and the result feeds into a cut list, homework check, or vector magnitude.

• • Skip the irrational arithmetic: The hypotenuse is a*sqrt(2) and the perimeter is a*(2+sqrt(2)). The page carries the sqrt(2) and the rounding, so the answer is reported to four decimals without an intermediate step.
• • Solve in both directions: Most problems give a leg and ask for the hypotenuse; some give a diagonal and ask for the square side. The Solve for selector handles both without switching pages.
• • Get the full 45-45-90 picture in one pass: The result panel shows the hypotenuse, leg, area, perimeter, and the 1 : 1 : sqrt(2) ratio, so a follow-up question does not need a second calculator.
• • Stay in the same unit end to end: The unit selector controls every output, including the area unit squared, so the answer matches the tape or ruler that produced the input.
• • Catch input mistakes early: The validator blocks zero, negative, or out-of-range side inputs, which is the kind of error that quietly turns a 7 cm cut into a 0.7 cm cut on a miter saw.

For a general isosceles triangle where the apex is not a right angle, the Isosceles Triangle page handles the two-equal-sides case with the law of cosines.

Three factors decide which value the page returns and how it is displayed, plus two limitations that keep the result honest.

• • The calculator only handles isosceles right triangles. A right triangle with two unequal legs needs the general Hypotenuse Calculator, and an isosceles triangle without a right angle needs the Isosceles Triangle solver.
• • Output is rounded to four decimals. The sqrt(2) factor is irrational, so the displayed hypotenuse and perimeter are close to but not exactly equal to the mathematical truth.

According to Wikipedia, Special right triangle, the 45-45-90 triangle has sides in the ratio 1 : 1 : sqrt(2), and Wolfram MathWorld, Pythagorean theorem confirms the Pythagorean identity reduces to A squared = 2 B squared, so A = B * sqrt(2).

When the area needs to come from the base and the equal side rather than from the two equal legs, the Isosceles Triangle Area Calculator page uses the standard isosceles area formula with the apex height.