Pythagorean Theorem Calculator

The Pythagorean theorem is a foundational geometric principle applying strictly to right triangles, which are triangles featuring one 90-degree angle. The rule states that the square of the length of the hypotenuse (the longest side, directly opposite the right angle) equals the sum of the squares of the lengths of the two shorter legs (a and b). Expressed algebraically, the formula is a² + b² = c², where c represents the hypotenuse. Additionally, the calculator computes the triangle's area using the formula Area = (a × b) / 2 and the perimeter by summing all three sides: Perimeter = a + b + c.

Imagine you are building a wooden ramp and need to verify your cuts. You measure leg a as 6 inches and leg b as 8 inches. To find the hypotenuse c, first square both leg lengths: 6 squared is 36, and 8 squared is 64. Next, add these squared values together: 36 + 64 = 100. Finally, take the square root of 100, which gives you a hypotenuse of exactly 10 inches. To find the area, multiply the legs (6 × 8 = 48) and divide by 2, resulting in an area of 24 square inches. The perimeter is simply the sum of all sides: 6 + 8 + 10 = 24 inches.

Always double-check that your triangle actually contains a 90-degree right angle before applying this theorem, as it does not work for acute or obtuse triangles. When dealing with square roots that do not yield whole numbers, decide in advance whether your project requires fractional or decimal rounding. Finally, remember that the hypotenuse must always be longer than either individual leg; if your calculated hypotenuse is shorter than a leg, recheck your input values.