Calculator Pythagorean Theorem

Calculate the unknown side of a right-angled triangle using the Pythagorean theorem (a² + b² = c²).

Result

Understanding the Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in Euclidean geometry that describes the relationship between the three sides of a right-angled triangle. A right-angled triangle is a triangle that contains one angle measuring exactly 90 degrees.

The Formula

The theorem states that the square of the length of the hypotenuse (the side opposite the right angle, usually denoted as 'c') is equal to the sum of the squares of the lengths of the other two sides, called legs (usually denoted as 'a' and 'b'). Mathematically, this is expressed as:

a² + b² = c²

How the Calculator Works

This calculator utilizes the Pythagorean theorem to find the length of an unknown side of a right-angled triangle. You can choose to calculate:

• The Hypotenuse (c): If you provide the lengths of the two legs (a and b), the calculator computes c = √(a² + b²).
• A Leg (a or b): If you provide the length of one leg and the hypotenuse, the calculator can find the other leg. For example, to find leg 'a', it uses the formula a = √(c² - b²). Similarly, for leg 'b', it uses b = √(c² - a²).

Use Cases

The Pythagorean theorem and this calculator have numerous practical applications in various fields:

• Construction and Architecture: Ensuring walls are perfectly perpendicular, calculating diagonal bracing, and determining roof pitches.
• Navigation: Calculating the shortest distance between two points on a map or the distance an aircraft has traveled.
• Surveying: Determining distances and boundaries in land measurement.
• Physics and Engineering: Calculating vector magnitudes and resultant forces.
• Computer Graphics: Determining distances for rendering and collision detection.
• Everyday Problem Solving: Figuring out if a large object will fit through a doorway diagonally, or calculating the length of a ladder needed to reach a certain height.

Example Calculations:

Scenario 1: Find the Hypotenuse

If leg 'a' is 3 units and leg 'b' is 4 units:

c² = 3² + 4² = 9 + 16 = 25

c = √25 = 5 units.

Scenario 2: Find Leg 'a'

If the hypotenuse 'c' is 13 units and leg 'b' is 5 units:

a² = c² - b² = 13² - 5² = 169 - 25 = 144

a = √144 = 12 units.