# How to Calculate the Diagonal of a Square: A Comprehensive Guide

**Introduction**

The diagonal of a square is a line segment running through two opposite corners of the shape, connecting them. Calculating the diagonal length is an essential skill in geometry, mathematics, and other practical applications such as construction or design. This article will guide you step-by-step on how to calculate the diagonal of a square using simple mathematical formulas.

**Understanding the Square**

A square is a four-sided polygon with equal sides and angles, also known as a quadrilateral. All internal angles in a square are 90 degrees (right angles), and the sides have equal lengths.

**Using Pythagorean Theorem**

To calculate the diagonal length of a square, you can use the Pythagorean theorem. The theorem states that, in any right-angled triangle, the sum of the squares of the two shorter sides’ lengths equals the square of the hypotenuse (the side opposite the right angle) in length.

In a square, when you draw one of its diagonals, it creates two right-angled triangles with equal sides. The diagonal itself becomes the hypotenuse of both triangles.

Here’s how to apply the Pythagorean theorem to find the diagonal:

**1. Determine the length of one side:** Measure or know one side’s length (s) of your square.

**2. Apply Pythagorean theorem formula:** Since both shorter sides are equal in length, you can use either one in your calculation:

Diagonal² = Side¹² + Side²²

Since both sides have equal lengths (s), you can rewrite this formula as:

Diagonal² = s² + s²

**3. Calculate and simplify:** Combine like terms:

Diagonal² = 2s²

**4. Find the diagonal:** To get the actual length of the diagonal, take the square root of your result:

Diagonal = √(2s²)

Example Calculation

Let’s calculate the diagonal of a square with side length 4 cm:

1. Determine the length of one side: s = 4 cm

2. Apply Pythagorean theorem formula:

Diagonal² = (4 cm)² + (4 cm)²

Diagonal² = 16 cm² + 16 cm²

Diagonal² = 32 cm²

3. Calculate and simplify:

Diagonal = √(32 cm²)

Diagonal ≈ 5.66 cm

So, the length of the diagonal is approximately 5.66 cm.

**Conclusion**

Calculating the diagonal of a square is quite simple and follows a straightforward formula based on the Pythagorean theorem. By understanding this fundamental geometric concept, you’ll be able to apply it across various mathematical, practical, or design-based disciplines. Always remember to consider units of measurement while working with real-life scenarios for accurate results.