Finding the Hypotenuse of an Isosceles Right Triangle
In an isosceles right triangle (45-45-90 triangle), the two legs are equal and the hypotenuse can be found using the simple relationship h = a * sqrt(2), where a is the length of either leg. This is derived directly from the Pythagorean theorem.
Step-by-Step Derivation
Step 1: Pythagorean Theorem
Start with the Pythagorean theorem: a² + b² = c². Since a = b in an isosceles right triangle:
Step 2: Simplify
Combine like terms on the left side.
Step 3: Solve for c
Take the square root of both sides.
Reverse: Solve for a
To find the leg from the hypotenuse, divide by sqrt(2).
Why This Relationship Matters
The ratio 1 : 1 : sqrt(2) is one of the most commonly encountered ratios in mathematics and applied sciences. Knowing that the hypotenuse of an isosceles right triangle is exactly sqrt(2) times the leg allows quick mental calculations in many situations.
Common Examples
- Leg = 1: Hypotenuse = sqrt(2) = 1.41421...
- Leg = 5: Hypotenuse = 5*sqrt(2) = 7.07107...
- Leg = 10: Hypotenuse = 10*sqrt(2) = 14.14214...
- Hypotenuse = 10: Leg = 10/sqrt(2) = 5*sqrt(2) = 7.07107...
- Hypotenuse = 1: Leg = sqrt(2)/2 = 0.70711...
Rationalizing the Denominator
When finding the leg from the hypotenuse, c/sqrt(2) can be rationalized by multiplying numerator and denominator by sqrt(2):
a = c / sqrt(2) = c * sqrt(2) / (sqrt(2) * sqrt(2)) = c * sqrt(2) / 2
Both forms are equivalent. The rationalized form (c * sqrt(2) / 2) is often preferred in exact calculations.
Diagonal of a Square
The diagonal of a square with side length s creates two isosceles right triangles, each with legs of length s. Therefore, the diagonal d = s * sqrt(2). This is the most common real-world application of this formula -- any time you cut a square diagonally, you create this triangle.
How to Use This Calculator
- Select whether you want to find the hypotenuse or a leg.
- Enter the known value.
- The calculator shows the result along with area, perimeter, altitude, and a complete step-by-step derivation.