How this instrument works
An isosceles right triangle has one right angle and two equal legs, which forces the remaining two angles to also be equal — each exactly 45°, since the three angles must sum to 180° and the two non-right angles must match. This is the classic 45-45-90 triangle, and because both legs are already known to be equal, the Pythagorean theorem, leg² + leg² = hypotenuse², simplifies directly to hypotenuse = leg × √2, needing only that single leg length as input.
The √2 factor is worth knowing on sight — it's one of the handful of fixed ratios (alongside the 30-60-90 triangle's √3) that show up constantly in geometry and trigonometry, since a 45-45-90 triangle is exactly half of a square, cut along its diagonal. The diagonal of any square is always its side length times √2, the identical relationship in different clothing.
This calculator deliberately takes only the leg as input, since the isosceles right triangle's shape leaves nothing else to specify — unlike a general right triangle, which genuinely needs two independent pieces of information (two legs, or a leg and an angle) to be pinned down completely.
- Enter the length of one of the two equal legs into the Leg length field.
- Read Hypotenuse: the sheet multiplies the leg length by √2 directly.
- Change the leg length to see the hypotenuse scale with it, always at that same fixed √2 ratio.
Worked example — legs of 5
An isosceles right triangle has both equal legs at 5. Its hypotenuse is 5√2 ≈ 7.07 — a direct application of the Pythagorean theorem, √(5²+5²) = √50 = 5√2, simplified down to the single-multiplication shortcut this special triangle allows.
A unit isosceles right triangle, with legs of exactly 1, has a hypotenuse of exactly √2 ≈ 1.414 — the reference ratio every larger isosceles right triangle's hypotenuse scales from. A larger triangle with legs of 10 has a hypotenuse of 10√2 ≈ 14.14, the identical √2 ratio scaled up tenfold.
Questions
What is the formula for an isosceles right triangle's hypotenuse?
hypotenuse = leg × √2, where leg is the length of either of the two equal legs. This comes directly from the Pythagorean theorem, simplified because both legs are already known to be identical.
Why are the two non-right angles always 45° each?
Because a triangle's three angles always sum to 180°, and one of them is already fixed at 90°, leaving 90° to split between the remaining two. Since the two legs are equal, those two remaining angles must also be equal, splitting that 90° evenly into 45° and 45°.
Why is √2 the specific ratio here?
An isosceles right triangle is exactly half of a square, cut along its diagonal. A square's diagonal is always its side length times √2, and that's the identical relationship appearing here between an isosceles right triangle's leg and its hypotenuse.
Do I need to enter both legs?
No — since both legs are equal by definition in an isosceles right triangle, only one needs to be entered. This is exactly what distinguishes this calculator from a general right-triangle solver, which needs two independent measurements.
How is this related to the 30-60-90 triangle?
Both are 'special right triangles' with fixed, memorizable side ratios — the 45-45-90 triangle's sides scale as 1:1:√2, while the 30-60-90 triangle's scale as 1:√3:2. Each arises from cutting a different regular shape (a square versus an equilateral triangle) in half.