Using the Pythagorean Theorem in Cuboids - Examples, Exercises and Solutions

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

Practice Using the Pythagorean Theorem in Cuboids

Question 1

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

Question 2

Look at the orthohedron below.

$$ D^1C^1=10 $$

$$ AA^1=12 $$

Calculate $$ A^1B $$ .

Question 3

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

Calculate the diagonal of the rectangular prism.

Question 4

Look at the orthohedron in the figure below.

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

Question 5

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

Examples with solutions for Using the Pythagorean Theorem in Cuboids

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

$\sqrt{65}$

Question 1

Look at the orthohedron in the figure below.

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

How long is the dotted line?

Question 2

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

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

Calculate the diagonal of the rectangular prism.

Question 3

Calculate the length of the dotted diagonal in the rectangular prism.

Question 4

Look at the box in the drawing and calculate the indicated diagonal.

Answer

$\sqrt{CD^2+FG^2+EA^2}$

Question 1

A cube has a side length of 5 cm.

Calculate the diagonal of the cube.

Question 2

A cuboid has a width measuring 8 cm and a height of 4 cm.

Calculate the length of the side AC.

Question 3

A rectangular prism has a height twice as long as its length.

Its width is 3 times greater than its height.

How many times greater is its diagonal than its length?

Question 4

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?

Question 5

Look at the orthohedron below.

$$ DC=DD_1 $$

What is the length of the orthohedron's diagonal?

Exercise #11

A cube has a side length of 5 cm.

Calculate the diagonal of the cube.

Video Solution

Answer

$5\sqrt{3}$

Exercise #12

A cuboid has a width measuring 8 cm and a height of 4 cm.

Calculate the length of the side AC.

Video Solution

Answer

$\sqrt{80}$ cm

Exercise #13

A rectangular prism has a height twice as long as its length.

Its width is 3 times greater than its height.

How many times greater is its diagonal than its length?

Video Solution

Answer

$\sqrt{41}$

Exercise #14

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?

Video Solution

Answer

You can, $36\cdot\sqrt{19}$ cm³

Exercise #15

Look at the orthohedron below.

$DC=DD_1$

What is the length of the orthohedron's diagonal?

Video Solution

Answer

$\sqrt{264}$

Topics learned in later sections

Congruence of Right Triangles (using the Pythagorean Theorem)

Using the Pythagorean Theorem in Cuboids - Examples, Exercises and Solutions

Pythagorean Theorem in an Orthohedron

Reminder of the Pythagorean theorem:

The Space Diagonal

Suggested Topics to Practice in Advance

Practice Using the Pythagorean Theorem in Cuboids

Examples with solutions for Using the Pythagorean Theorem in Cuboids

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Answer

Exercise #5

Video Solution

Answer

Exercise #6

Video Solution

Answer

Exercise #7

Video Solution

Answer

Exercise #8

Video Solution

Answer

Exercise #9

Video Solution

Answer

Exercise #10

Video Solution

Answer

Exercise #11

Video Solution

Answer

Exercise #12

Video Solution

Answer

Exercise #13

Video Solution

Answer

Exercise #14

Video Solution

Answer

Exercise #15

Video Solution

Answer

Topics learned in later sections