Question

What is the longest line segment that can be drawn in a right rectangular prism that is 12 cm​ long, 11 cm​ wide, and 8 cm​ tall?

Answer 1

To find the longest line segment that can be drawn in a right rectangular prism (also called a box), we need to calculate the length of the space diagonal of the prism. The formula to find the length of the space diagonal \( d \) of a rectangular prism with dimensions \( l \) (length), \( w \) (width), and \( h \) (height) is:

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

In this case, the dimensions are:
- Length \( l = 12 \, \text{cm} \)
- Width \( w = 11 \, \text{cm} \)
- Height \( h = 8 \, \text{cm} \)

Plugging in the dimensions into the formula:

\[
d = \sqrt{(12)^2 + (11)^2 + (8)^2}
\]

Calculating each term:

\[
(12)^2 = 144
\]
\[
(11)^2 = 121
\]
\[
(8)^2 = 64
\]

Now, add these values together:

\[
144 + 121 + 64 = 329
\]

Now, take the square root:

\[
d = \sqrt{329}
\]

Calculating the square root:

\[
d \approx 18.14 \, \text{cm}
\]

Therefore, the longest line segment that can be drawn in the right rectangular prism is approximately **18.14 cm**.