Given the cuboid whose length is equal to 9 cm Width is equal to 3 cm Side AB equals 10 cm Is it possible to calculate the volume of the cuboid? 9 9 9 3 3 3 10 10 10 A A A B B B

You can, $$ 36\cdot\sqrt{19} $$ cm³

The Application of the Pythagorean Theorem to an Orthohedron or Cuboid

Pythagorean Theorem in an Orthohedron

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.

Reminder of the Pythagorean theorem:

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Basiclly:
$a²+b² = c²$
This principle can be extended to three-dimensional shapes like cuboids or orthohedrons.

Diagram of a rectangular prism (cuboid) illustrating its diagonal, labeled 'Diagonal del ortoedro.' The orange diagonal highlights the use of the Pythagorean theorem to calculate its length. Featured in a guide on applying the Pythagorean theorem to 3D shapes.

The Space Diagonal

The Pythagorean theorem can help find the length of the diagonals on the faces of an orthohedron, but it also extends to finding the space diagonal—the diagonal that runs through the interior of a cuboid from one corner to its opposite corner.

There are two methods to find this:

Use the Pythagorean theorem twice: First, find one face diagonal, then use it to calculate the space diagonal.
Use the formula:
$d=l^2+w^2+h^2$
where $I$ , $w$ , and $h$ are the length, width, and height of the cuboid, respectively.

Detailed explanation

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Test yourself on using the pythagorean theorem in cuboids!

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

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

Video Solution

Step-by-Step Solution

We will use the Pythagorean theorem to find diagonal AD1:

$AA_1+A_1D_1=AD_1$

Let's input the known data:

$5^2+7^2=D_1A^2$

$D_1A^2=25+49=74$

Let's find the square root:

$AD_1=\sqrt{74}$

From the data we can see that:

$AA_1=DD_1=5$

Now let's look at triangle DD1C1 and calculate DC1 using the Pythagorean theorem:

$D_1D^2+D_1C_1^2=C_1D^2$

Let's input the existing data:

$5^2+4^2=C_1D^2$

$C_1D^2=25+16=41$

Let's find the square root:

$DC_1=\sqrt{41}$

Now let's focus on triangle A1D1C1 and find diagonal A1C1:

$A_1D_1^2+D_1C_1^2=A_1C_1^2$

Let's input the known data:

$7^2+4^2=A_1C_1^2$

$A_1C_1^2=49+16=65$

Let's find the square root:

$A_1C_1=\sqrt{65}$

Now we have all 3 lengths of all possible diagonal corners in the box:

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

Answer

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

Exercise #2

Look at the orthohedron below.

$D^1C^1=10$

$AA^1=12$

Calculate $A^1B$ .

Video Solution

Step-by-Step Solution

From the given data, we know that:

$D_1C_1=A_1B_1=AB=10$

Let's draw a diagonal between A1 and B and focus on triangle AA1B.

We'll calculate A1B using the Pythagorean theorem:

$AA_1^2+AB^2=A_1B^2$

Then we will substitute in the known values:

$12^2+10^2=A_1B^2$

$A_1B^2=144+100=244$

Finally, we calculate square root:

$A_1B=\sqrt{244}$

$A_1B=\sqrt{4\times61}=\sqrt{4}\times\sqrt{61}$

$A_1B=2\sqrt{61}$

Answer

$2\sqrt{61}$

Exercise #3

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 #4

Look at the orthohedron in the figure below.

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

Video Solution

Answer

$HBE$

Exercise #5

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

Video Solution

Answer

$\sqrt{65}$

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Test your knowledge

Question 1

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

Question 2

Look at the orthohedron in the figure below.

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

How long is the dotted line?

Question 3

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

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

Calculate the diagonal of the rectangular prism.

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