Calculate the lengths of all possible diagonals on the faces of the rectangular prism below: 4 4 4 7 7 7 5 5 5

$$ \sqrt{74},\sqrt{41},\sqrt{65} $$

$$ \sqrt{74},\sqrt{24},\sqrt{65} $$

The Application of the Pythagorean Theorem to an Orthohedron or Cuboid

The orthohedron or cuboid is a rectangular prism, a three-dimensional figure, that is, it has length, width, and height (or depth). In addition, the angles between the different planes are right angles, which allows us to make use of the Pythagorean theorem to calculate the length of different sections of the orthohedron.

A1- Uses of the Pythagorean theorem in an orthohedron

Detailed explanation

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Test yourself on use of the pythagorean theorem in the orthohedron!

Look at the orthohedron in the figure below.

Which angle is between the diagonal BH and the face ABFE?

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We will illustrate this with an example.

Given an orthohedron as represented in the diagram.

The dimensions of the box are $6$ , $8$ and $10$ .

We are asked to calculate the dimensions of the diagonal of the lower base of the box.

We will look at the diagram and see that the base of the box is, in fact, a rectangle whose edges measure $6$ and $8$ . These edges also serve as legs with a right angle between them.

Therefore, we will use the Pythagorean theorem and calculate the hypotenuse which, in fact, is the required diagonal.

According to the Pythagorean theorem we will obtain:

$X=10$

That is, the diagonal measures $10$ .

Uses of the Pythagorean theorem in an orthohedron

If you are interested in learning more about other triangle topics, you can go to one of the following articles:

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Examples and exercises with solutions on the application of the Pythagorean theorem in a rectangular prism or cuboid

Exercise #1

Shown below is the rectangular prism $ABCDA^1B^1C^1D^1$ .

Calculate the diagonal of the rectangular prism.

Video Solution

Step-by-Step Solution

Let's look at face CC1D1D and use the Pythagorean theorem to find the diagonal of the face:

$D_1C_1^2+CC_1^2=D_1C^2$

Let's insert the known data:

$10^2+4^2=D_1C^2$

$116=D_1C^2$

Let's focus a bit on triangle BCD1 and use the Pythagorean theorem to find diagonal BD1:

$D_1C^2+CB^2=BD_1^2$

Let's insert the known data:

$116+7^2=BD_1^2$

$116+49=BD_1^2$

$165=BD_1^2$

Let's find the root:

$\sqrt{165}=BD_1$

Answer

$\sqrt{165}$

Exercise #2

Look at the orthohedron in the figure below.

Which angle is between the diagonal BH and the face ABFE?

Video Solution

Answer

$HBE$

Exercise #3

$ABCDA^1B^1C^1D^1$ is a rectangular prism.

$AB=7$
$AA^1=5$

Calculate the diagonal of the rectangular prism.

Video Solution

Answer

Not enough data

Exercise #4

Look at the orthohedron in the figure and calculate the length of the dotted line.

Video Solution

Answer

$\sqrt{65}$

Exercise #5

Look at the orthohedron in the figure below.

$DCC^1D^1$ is a square.

How long is the dotted line?

Video Solution

Answer

$13$

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Test your knowledge

Question 1

$$ ABCDA^1B^1C^1D^1 $$ is a rectangular prism.

$$ AB=7 $$
$$ AA^1=5 $$

Calculate the diagonal of the rectangular prism.

Question 2

Look at the orthohedron in the figure and calculate the length of the dotted line.

Question 3

Look at the orthohedron in the figure below.

$$ DCC^1D^1 $$ is a square.

How long is the dotted line?

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