Pythagorean Theorem

Overview

Engineering reference data for Pythagorean Theorem in miscellaneous.

Key Formulas

Unit Conversion

$y = x \cdot k$

Multiply by conversion factor.

Linear Interpolation

$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$

Estimate between two known points.

Percentage

$p = \frac{\text{part}}{\text{whole}} \times 100\%$

Part as fraction of whole.

Variables

The Sine Rule

The Sine Rule relates the sides and angles of any triangle:

$\frac{A}{\sin(a)} = \frac{B}{\sin(b)} = \frac{C}{\sin(c)}$

where $A, B, C$ are the side lengths opposite to their respective angles $a, b, c$. This rule is essential for solving oblique triangles (non-right triangles).

Verifying a Square Corner

A common engineering application of the Pythagorean theorem is to verify if a corner is square (90 degrees). This is often done using the 3-4-5 triangle rule:

• If $A = 3$ units, $B = 4$ units, and $C = 5$ units (where $C$ is the hypotenuse), then the angle between sides $A$ and $B$ is exactly 90 degrees.
• This principle is scalable: any triangle with side lengths in the ratio 3:4:5 will be a right triangle.

The relationship holds because $3^2 + 4^2 = 5^2$, satisfying the Pythagorean theorem. This method provides a practical, measurable check for squareness in construction and layout work.

References

• Original Source