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# A rectangular prism has a length of 8, width of 9, and height of 12. What is the length of its longest internal diagonal?

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A rectangular prism has a length of 8, width of 9, and height of 12. What is the length of its longest internal diagonal?

Sep 16, 2017

#1
+7352
+1

The longest diagonal will be from one corner to the corner completely opposite it (top and bottom, front and back, left and right). Like the black line in this picture:

We can find it's length by making it into a triangle like this:

Let's call this new gray line  " b ".

Now we can use the Pythagorean theorem to find  b  and  the length of the prism's diagonal.

92 + 82  =  b2

81 + 64  =  b2

145  =  b2

b2 + 122  =  (diagonal)2                 Substitute  145  in for  b2  .

145 + 122  =  (diagonal)2

145 + 144  =  (diagonal)2

289  =  (diagonal)2              Take the positive square root of both sides.

√289  =  diagonal

17  =  diagonal

Sep 16, 2017

#1
+7352
+1

The longest diagonal will be from one corner to the corner completely opposite it (top and bottom, front and back, left and right). Like the black line in this picture:

We can find it's length by making it into a triangle like this:

Let's call this new gray line  " b ".

Now we can use the Pythagorean theorem to find  b  and  the length of the prism's diagonal.

92 + 82  =  b2

81 + 64  =  b2

145  =  b2

b2 + 122  =  (diagonal)2                 Substitute  145  in for  b2  .

145 + 122  =  (diagonal)2

145 + 144  =  (diagonal)2

289  =  (diagonal)2              Take the positive square root of both sides.

√289  =  diagonal

17  =  diagonal

hectictar Sep 16, 2017
#2
+98173
+1

Nice diagram, hectictar....!!!

Sep 16, 2017