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# Find The Diagonal Of The Rectangular Solid, Diagonal Of A Rectangle

Intermediate Geometry Help » Solid Geometry » Prisms » How to find the diagonal of a prism

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What is the length of the diagonal of a rectangular box with the dimensions of

?

Explanation:

To solve this problem we need an extension of the Pythagorean Theorem:

So the equation to solve becomes

So the distance of the diagonal is

.

Explanation:

The length of the diagonal is from the bottom left hand corner closest to us to the top right hand corner that”s farthest away from us.

This kind of a problem may seem to be a little more complicated than it really is.

In order to solve for the diagonal length, all that”s required is the Pythagorean Theorem. This equation will be used twice to solve for the dashed line.

For the first step of this problem, it”s helpful to imagine a triangle “slice” that”s being taken inside the prism.

, where the diagonal of interest is D2, and D1 is the diagonal that cuts from corner to corner of the bottom face of the prism. Of this triangle that”s outlined in pink dashed lines, the given information (the dimensions of the prism) provides a length for one of the legs (16).

We can already “map out” that D2 (the hypotenuse of the dashed triangle) can be solved by using the Pythagorean Theorem if we can obtain the length of the other leg (D1).

The next step of this problem is to solve for D1. This will be the first use of the Pythagorean theorem. D1 is the diagonal of the base and is limited to a 2D face. This can be represented as:

The hypotenuse of the base, or the mystery length leg of the dashed triangle, can be solved by using the Pythagorean Theorem:

Now that we calculated the length of D1, D2 can be solved for by using the Pythagorean Theorem a second time: