Find the hypotenuse of an isosceles right triangle from its leg, or find the leg from the hypotenuse. Includes step-by-step solution, all triangle properties, visual bars, and a reference table.
The hypotenuse of an isosceles right triangle has a beautifully simple relationship with the legs: it equals the leg length times the square root of 2. This single formula, c = a√2, is one of the most frequently used relationships in geometry, construction, and standardized testing.
This calculator focuses on that core conversion—leg to hypotenuse and hypotenuse to leg—while also computing every other property of the triangle: area, perimeter, height to the hypotenuse, inradius, circumradius, and median lengths. An optional step-by-step display walks you through each calculation so you can follow (or teach) the process.
Why a dedicated hypotenuse tool? Because the question "what is the hypotenuse of a 45-45-90 triangle with leg X?" appears constantly—in homework, on the SAT/ACT, in carpentry when computing diagonal braces, and in tiling when cutting squares diagonally. Having instant, precise answers with full context saves time and prevents rounding errors.
Eight presets cover the most common leg and hypotenuse values, a bar chart visualises all dimensions relative to the hypotenuse, and a reference table lists ten leg-hypotenuse pairs for quick lookup. Select your unit, adjust decimal precision, and toggle the step-by-step view to suit your needs.
Isosceles Right Triangle Hypotenuse problems often require several dependent steps, and a small arithmetic slip can propagate through every derived value. This calculator is tailored to that workflow: you enter decimal places, known value, unit, and it returns hypotenuse, leg, area, perimeter in one consistent pass. It is useful for homework checks, worksheet generation, tutoring walkthroughs, and fast field/design estimates where you need reliable geometry results without rebuilding the full derivation each time.
Hypotenuse = leg × √2. Leg = hypotenuse / √2. Area = leg² / 2. Perimeter = 2·leg + hypotenuse. Height to hypotenuse = leg / √2. Circumradius = hypotenuse / 2.
Result: Hypotenuse ≈ 14.1421 cm
Hypotenuse = 10 × √2 = 10 × 1.41421… ≈ 14.1421 cm. Area = 100/2 = 50 cm². Perimeter ≈ 34.1421 cm.
This isosceles right triangle hypotenuse tool links the entered values (decimal places, known value, unit, show steps) to the target geometry relationships used in class and practice problems. Instead of solving each intermediate step manually, you can validate setup and arithmetic quickly while still tracing which measurements drive the final result.
Formula focus: the calculator formula
Isosceles Right Triangle Hypotenuse shows up in school geometry, technical drafting, construction layout checks, and early engineering design estimates. When values are changed repeatedly, the calculator helps you compare scenarios quickly and see how sensitive the shape is to each dimension.
Start with the primary outputs (hypotenuse, leg, area, perimeter) and then use the remaining cards/tables to confirm consistency with your diagram. Keep units consistent across inputs, and round only at the end if your assignment or project specifies a fixed precision.
Hypotenuse = leg × √2. So if the leg is 10, the hypotenuse is 10√2 ≈ 14.142.
Divide the hypotenuse by √2 (or multiply by √2/2). For example, hypotenuse 14.142 → leg ≈ 10.
By the Pythagorean theorem, c² = a² + a² = 2a², so c = a√2. Use this as a practical reminder before finalizing the result.
If the leg is a rational number, the hypotenuse is irrational because √2 is irrational. Integer hypotenuses require irrational legs.
It equals the leg divided by √2 (= a/√2 = a√2/2), which is also half the hypotenuse divided by… well, it simplifies to a/√2. Keep this note short and outcome-focused for reuse.
This tool is focused specifically on the leg ↔ hypotenuse conversion with an optional step-by-step walkthrough, targeting the most common search query for this triangle type. Apply this check where your workflow is most sensitive.