Finding the diagonal of a right rectangular prism
Formula for the diagonal in terms of the length, width, and height of the rectangular prism
The diagonal of a right rectangular prism goes from one corner of the prism, across the interior volume, all the way to the opposite corner of the prism.
You can find the length of a diagonal of a right rectangular prism using
???d=\sqrt{{{l}^{2}}+{{w}^{2}}+{{h}^{2}}}???
where ???d??? is the length of the diagonal, and ???l???, ???w???, and ???h??? are the length, width, and height, respectively.
How to find the length of the diagonal
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Length of the diagonal of a right rectangular prism
Example
What is the length of the diagonal of the right rectangular prism?
Not all of the dimensions here are the same. Change ???60\text{ mm}??? to centimeters first.
???60\text{ mm}=6\text{ cm}???
Then the dimensions are
Plugging these into the formula for the diagonal, we get
???d=\sqrt{{{l}^{2}}+{{w}^{2}}+{{h}^{2}}}???
???d=\sqrt{{{7}^{2}}+{{12}^{2}}+{{6}^{2}}}???
???d=\sqrt{49+144+36}???
???d=\sqrt{229}??? cm
Let’s try another one.
Example
Find the width of the right rectangular prism.
We just need to plug the dimensions we’ve been given into the formula for the diagonal.
???d=\sqrt{{{l}^{2}}+{{w}^{2}}+{{h}^{2}}}???
???7\sqrt{2}=\sqrt{{{5}^{2}}+{{w}^{2}}+{{3}^{2}}}???
Manipulate the equation to solve for ???w???.
???(49)(2)={{5}^{2}}+{{w}^{2}}+{{3}^{2}}???
???98=25+{{w}^{2}}+9???
???98=34+{{w}^{2}}???
???64={{w}^{2}}???
???8=w???