What is the length of the space diagonal in a cuboid with dimensions 6 cm, 8 cm, and 10 cm?

The length of the space diagonal in a cuboid with dimensions 6 cm, 8 cm, and 10 cm is 14 cm.

To find the length of the space diagonal in a cuboid, we use the formula derived from the Pythagorean theorem in three dimensions. The formula is:

\[ d = \sqrt{l^2 + w^2 + h^2} \]

where \( d \) is the space diagonal, \( l \) is the length, \( w \) is the width, and \( h \) is the height of the cuboid. For our cuboid, the dimensions are 6 cm (length), 8 cm (width), and 10 cm (height).

First, we square each of the dimensions:
\[ 6^2 = 36 \]
\[ 8^2 = 64 \]
\[ 10^2 = 100 \]

Next, we add these squared values together:
\[ 36 + 64 + 100 = 200 \]

Finally, we take the square root of the sum to find the length of the space diagonal:
\[ d = \sqrt{200} \]

To simplify \(\sqrt{200}\), we can break it down:
\[ \sqrt{200} = \sqrt{4 \times 50} = \sqrt{4} \times \sqrt{50} = 2 \times \sqrt{50} \]

Since \(\sqrt{50}\) is approximately 7.07, we get:
\[ d \approx 2 \times 7.07 = 14.14 \]

Rounding to the nearest whole number, the length of the space diagonal is approximately 14 cm. This method ensures you understand how to apply the Pythagorean theorem in three dimensions to solve for the space diagonal in any cuboid.