Question

How do you find the diagonal of a rectangular prism if it has a length of 12cm, 9cm width, and 8cm height?

Answer 1

Hint: In this question we have to find the length of the diagonal of the rectangular prism which is given by formula, \[{d^2} = {l^2} + {b^2} + {h^2}\] or $$d = \sqrt {{l^2} + {b^2} + {h^2}} $$, where\[l\]is the length, $$b$$ is the width, and $$w$$ is the height of the prism, now substitute the given values in the formula, we will get the required length of the diagonal.

Complete step-by-step answer:
A rectangular prism can be defined as a 3-dimensional solid shape which has six faces that are rectangles.
The length of the diagonal is the square root of the sum of squares of length, width and height of the rectangular prism, which is given by formula,
 $${d^2} = {l^2} + {b^2} + {w^2}$$ or\[d = \sqrt {{l^2} + {b^2} + {w^2}} \], where$$l$$is the length, ..is the width, and $$w$$is the height of the prism.
So, here,\[l = 12\],
\[b = 9\],
$$h = 8$$,
Now substituting the values in the formula we get,
$${d^2} = {l^2} + {b^2} + {w^2}$$,
Now substituting ten values we get,
\[ \Rightarrow {d^2} = {12^2} + {9^2} + {8^2}\],
Now squaring the values we get,
$$ \Rightarrow $$$${d^2} = 144 + 81 + 64$$,
Now simplifying we get,
\[ \Rightarrow \]$${d^2} = 289$$,
Now taking the square root we get,
$$d = \sqrt {289} $$,
Now simplifying we get,
$$ \Rightarrow $$$$d = 17$$,
The length of the diagonal is 17cm.

The length of the diagonal of the rectangular prism with length of 12cm, 9cm width, and 8cm height is 17cm.

Note:
 A rectangular prism (also cuboid, rectangular box, right rectangular prism) is a three dimensional figure with 6 faces that are all rectangles.
Here are some useful formulas related to rectangular prism:
The volume can be determined by multiplying the length, width, and height, $$V = lbh$$.
The length of the diagonals can be determined by using the formula\[{d^2} = {l^2} + {b^2} + {h^2}\].
The surface area of the prism is $$2lb + 2bh + 2lh$$.