How do you determine the length of the diagonal in a 3D rectangle with sides 5 cm, 12 cm, and 13 cm?

To determine the length of the diagonal in a 3D rectangle, use the 3D Pythagorean theorem.

In a 3D rectangle (also known as a cuboid), the length of the diagonal can be found using the 3D Pythagorean theorem. This theorem is an extension of the Pythagorean theorem used in 2D geometry. For a cuboid with side lengths \(a\), \(b\), and \(c\), the formula for the diagonal \(d\) is:

\[ d = \sqrt{a^2 + b^2 + c^2} \]

In this case, the sides of the cuboid are 5 cm, 12 cm, and 13 cm. Let's assign these values to \(a\), \(b\), and \(c\):

\[ a = 5 \, \text{cm} \]
\[ b = 12 \, \text{cm} \]
\[ c = 13 \, \text{cm} \]

Now, substitute these values into the formula:

\[ d = \sqrt{5^2 + 12^2 + 13^2} \]

Calculate the squares of each side:

\[ 5^2 = 25 \]
\[ 12^2 = 144 \]
\[ 13^2 = 169 \]

Add these values together:

\[ 25 + 144 + 169 = 338 \]

Finally, take the square root of the sum:

\[ d = \sqrt{338} \]

Using a calculator, you find:

\[ d \approx 18.39 \, \text{cm} \]

So, the length of the diagonal in the 3D rectangle is approximately 18.39 cm. This method ensures you accurately determine the diagonal length by considering all three dimensions of the cuboid.