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To understand how the diagonal of a square is calculated, we first need to explore the Pythagorean theorem. This fundamental principle in geometry states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
The formula is: \[ a^2 + b^2 = c^2 \] where \( a \) and \( b \) are the lengths of the legs, and \( c \) is the length of the hypotenuse.
In the context of a square, if you split it diagonally, you create two right-angled triangles. The two sides of each triangle are equal in length and represent the sides of the square.

Squares have fascinating properties that make geometry problems easy to solve once you understand them. Here are a few essential properties:

• All four sides of a square are of equal length.
• All four angles of a square measure 90 degrees.
• The diagonals of a square are equal in length and bisect each other at right angles.
• The diagonals divide the square into two identical right-angled triangles.

By knowing these properties, you can easily apply geometric principles to solve various problems. For instance, because each side of the square is equal, finding the diagonal can be straightforward using the Pythagorean theorem.

In geometry, certain formulas help simplify the process of solving problems. Here’s how you can find the diagonal of a square using a simple formula:

The formula for finding the diagonal \( d \) of a square with side length \( s \) is: \[ d = s \times \sqrt{2} \] This formula is derived from the Pythagorean theorem.
For example, in the original problem where each side of the square is 4 inches and applying the formula:
\[ d = 4 \times \sqrt{2} \]
Substituting the value of \( \sqrt{2} \) which is approximately 1.414, you get the diagonal as:
\[ 4 \times 1.414 \] which equals roughly 5.656 inches.
Understanding and applying these geometry formulas can make solving math problems more efficient and less intimidating.