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We use the Pythagorean theorem to find the length of a diagonal in a square. The Pythagorean theorem helps us relate the lengths of the sides of a right-angled triangle. In any right-angled triangle, the theorem states that the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse). The formula is written as: \[ a^2 + b^2 = c^2 \] where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the length of the hypotenuse. In our example, the sides of the square are 4 inches long. As the diagonal divides the square into two right-angled triangles, we use the sides of the square (4 inches) to apply the theorem. So, it translates to: \[ 4^2 + 4^2 = c^2 \].

A right-angled triangle is a type of triangle that has one angle exactly at 90 degrees. This means one angle forms a perfect L-shape. The side opposite this right angle is called the hypotenuse, and it is always the longest side of the triangle. The other two sides are known as legs. In the square diagonal problem, each diagonal of the square splits it into two right-angled triangles. Here, the legs of the triangles are the sides of the square, both measuring 4 inches. Therefore, in our equation, these are our \(a\) and \(b\) values.

Once we have set up our equation using the Pythagorean theorem, the next step is to solve for the hypotenuse. This involves calculating the square root. The square root is a value that, when multiplied by itself, gives the original number. In our problem, after substituting and squaring the sides, we combine them to get: \[16 + 16 = 32\]. Now, to solve for the diagonal \(c\), we take the square root of 32: \[c = \sqrt{32}\]. Furthermore, simplifying \(\sqrt{32}\), we can break it down into \(\sqrt{16 \cdot 2}\), which gives \(4 \sqrt{2}\). So, the length of the diagonal is \(4 \sqrt{2}\) inches.