Understanding the Pythagorean Theorem
The **Pythagorean theorem** is one of the most fundamental principles in geometry. Discovered by the ancient Greek mathematician Pythagoras, it states that in any right-angled triangle, the sum of the squares of the lengths of the legs (a and b) is equal to the square of the length of the hypotenuse (c), which is the side opposite the right angle.
The Core Equation
a² + b² = c²
This equation can be rearranged depending on which side length is unknown:
- Solving for Hypotenuse c: c = √(a² + b²)
- Solving for Leg a: a = √(c² - b²)
- Solving for Leg b: b = √(c² - a²)
Calculating Right Triangle Details
Beyond finding side lengths, a right triangle has several distinct geometric properties that our engine computes:
- Area: Since the legs a and b are perpendicular, they act as base and height, meaning: Area = 0.5 × a × b.
- Perimeter: The total length of the boundary: Perimeter = a + b + c.
- Altitude to Hypotenuse (h_c): The shortest distance from the right angle to the hypotenuse: h_c = (a × b) / c.
- Trigonometric Angles: Calculated in degrees using inverse trigonometry functions: α = arcsin(a / c) × (180 / π) and β = arcsin(b / c) × (180 / π).