# Use of the Pythagorean Theorem in the orthohedron - Examples, Exercises and Solutions

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.

### Suggested Topics to Practice in Advance

1. The Pythagorean Theorem

## Practice Use of the Pythagorean Theorem in the orthohedron

### Exercise #1

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

Calculate the diagonal of the rectangular prism.

### 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:

$\sqrt{165}=BD_1$

$\sqrt{165}$

### Exercise #2

Look at the orthohedron in the figure below.

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

### Video Solution

$HBE$

### Exercise #3

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

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

Calculate the diagonal of the rectangular prism.

Not enough data

### Exercise #4

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

### Video Solution

$\sqrt{65}$

### Exercise #5

Look at the orthohedron in the figure below.

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

How long is the dotted line?

### Video Solution

$13$

### Exercise #1

Look at the orthohedron below.

$D^1C^1=10$

$AA^1=12$

Calculate $A^1B$.

### Video Solution

$2\sqrt{61}$

### Exercise #2

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

### Video Solution

$\sqrt{113}$

### Exercise #3

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

### Video Solution

$x\sqrt{10}$

### Exercise #4

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

### Video Solution

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

### Exercise #5

Look at the rectangular prism in the figure and express the length of the diagonal using the sides $EA,CD,FG$.

### Video Solution

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

### Exercise #1

The side BC in the rectangular prism below is 8 cm long.

Side BD is 4 cm long.
Side AD is 5 cm long.

Calculate the volume of the cube.

96 cm³

### Exercise #2

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

You can, $36\cdot\sqrt{19}$ cm³

### Exercise #3

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

Calculate the length of the side AC.

### Video Solution

$\sqrt{80}$ cm

### Exercise #4

Look at the rectangular prism below.

$ABCDA^1B^1C^1D^1$

$BC^1=12$

$D^1D=10$

$D^1C=7$

### Video Solution

$2\sqrt{11}$ cm

### Exercise #5

Look at the rectangular prism in the figure below.

$2\sqrt{22}$ cm