Using the Pythagorean Theorem in Cuboids: Using Pythagoras' theorem to calculate the height of a box

Determining the size of the diagonal using Pythagoras' theorem Using Pythagoras' theorem- with parameters to calculate the lines of the box

Question 1

Look at the rectangular prism in the figure below.

The length of the diagonal is $2\sqrt{14}$ cm.

Calculate $$ BB^1 $$ .

Question 2

Look at the rectangular prism in the figure below.

Calculate AD.

Question 3

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

$$ D^1C^1=14 $$

$$ BD=20 $$

Calculate the height of the rectangular prism.

Question 4

Look at the orthohedron in the figure.

$$ A^1B=12 $$

Canculate $$ CC^1 $$ .

Question 5

Look at the rectangular prism below.

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

$$ BC^1=12 $$

$$ D^1D=10 $$

$$ D^1C=7 $$

Calculate AD.

Examples with solutions for Using the Pythagorean Theorem in Cuboids: Using Pythagoras' theorem to calculate the height of a box

Exercise #1

Look at the rectangular prism in the figure below.

The length of the diagonal is $2\sqrt{14}$ cm.

Calculate $BB^1$ .

Video Solution

Step-by-Step Solution

The problem asks us to find the height of a rectangular prism. Given the space diagonal $d = 2\sqrt{14}$ cm and base dimensions (length = 6 cm, width = 4 cm), we can use the formula for the space diagonal:

$d = \sqrt{l^2 + w^2 + h^2}$

Substitute the known values:

$2\sqrt{14} = \sqrt{6^2 + 4^2 + h^2}$

First, calculate $6^2 + 4^2$ :

$6^2 = 36, \quad 4^2 = 16$

Thus, $6^2 + 4^2 = 52$ .

Substitute back into the equation:

$(2\sqrt{14})^2 = 52 + h^2$

Calculate $(2\sqrt{14})^2$ :

$2^2 \times (\sqrt{14})^2 = 4 \times 14 = 56$

Substitute and solve for $h^2$ :

$56 = 52 + h^2$

Subtract 52 from both sides:

$h^2 = 4$

Take the square root of both sides:

$h = \sqrt{4} = 2$

Therefore, the height of the rectangular prism $BB^1$ is $2$ cm.

Answer

$2$

Exercise #2

Look at the rectangular prism in the figure below.

Calculate AD.

Video Solution

Answer

$2\sqrt{22}$ cm

Exercise #3

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

$D^1C^1=14$

$BD=20$

Calculate the height of the rectangular prism.

Video Solution

Answer

$2\sqrt{51}$ cm

Exercise #4

Look at the orthohedron in the figure.

$A^1B=12$

Canculate $CC^1$ .

Video Solution

Answer

$2\sqrt{11}$

Exercise #5

Look at the rectangular prism below.

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

$BC^1=12$

$D^1D=10$

$D^1C=7$

Calculate AD.

Video Solution

Answer

$2\sqrt{11}$ cm

Question 1

Look at the orthohedron in the figure below.

Calculate $$ DE $$ .

Question 2

Look at the rectangular prism in the figure below.

Calculate the length of $$ DD^1 $$ .

Question 3

$$ ABCDA^1B^1C^1D^1 $$ is an orthohedron.

$$ BC^1=2x $$

$$ A^1D^1=3y $$

Express $$ BB^1 $$ in terms of x and y.

Question 4

Look at the rectangular prism below and calculate $$ BC $$ .

Question 5

A rectangular prism has a diagonal length of $2\sqrt{30}$ cm.

It is $$ 4 $$ cm high and $$ 2 $$ cm wide.

Calculate the length of the rectangular prism.

Exercise #6

Look at the orthohedron in the figure below.

Calculate $DE$ .

Video Solution

Answer

It is not possible to calculate.

Exercise #7

Look at the rectangular prism in the figure below.

Calculate the length of $DD^1$ .

Video Solution

Answer

$6\sqrt{2}$ cm

Exercise #8

$ABCDA^1B^1C^1D^1$ is an orthohedron.

$BC^1=2x$

$A^1D^1=3y$

Express $BB^1$ in terms of x and y.

Video Solution

Answer

$\sqrt{4x^2-9y^2}$

Exercise #9

Look at the rectangular prism below and calculate $BC$ .

Video Solution

Answer

$2\sqrt{145}$

Exercise #10

A rectangular prism has a diagonal length of $2\sqrt{30}$ cm.

It is $4$ cm high and $2$ cm wide.

Calculate the length of the rectangular prism.

Video Solution

Answer

$10$ cm

Question 1

Look at the orthohedron in the figure below.

$$ BC^1=2AA^1 $$
$$ A^1D^1=5 $$

Calculate $$ AA^1 $$ .

Question 2

Look at the rectangular prism in the figure below and calculate $$ CC^1 $$ .

Question 3

Look at the rectangular prism in the figure below.

The diagonal is $$ 13x $$ long.

$$ A^1D^1=2y $$

$$ A^1B^1=2x $$

Calculate $$ DD^1 $$ .

Question 4

A rectangular prism has a diagonal length of 24 cm.

The surface area of one of the faces is 40 cm² and its side measures 4 cm.

Calculate the dimensions of the rectangular prism.

Question 5

Look at the orthohedron below.

$$ B^1C^1=7x $$

$$ DC=4x $$

$BD^1=2\sqrt{26}x$

Calculate $$ AA^1 $$ .

Exercise #11

Look at the orthohedron in the figure below.

$BC^1=2AA^1$
$A^1D^1=5$

Calculate $AA^1$ .

Video Solution

Answer

$\frac{5}{\sqrt{3}}$ cm

Exercise #12

Look at the rectangular prism in the figure below and calculate $CC^1$ .

Video Solution

Answer

$\sqrt{139}$ cm

Exercise #13

Look at the rectangular prism in the figure below.

The diagonal is $13x$ long.

$A^1D^1=2y$

$A^1B^1=2x$

Calculate $DD^1$ .

Video Solution

Answer

$\sqrt{165x^2-4y^2}$

Exercise #14

A rectangular prism has a diagonal length of 24 cm.

The surface area of one of the faces is 40 cm² and its side measures 4 cm.

Calculate the dimensions of the rectangular prism.

Video Solution

Answer

$10,4,2\sqrt{115}$

Exercise #15

Look at the orthohedron below.

$B^1C^1=7x$

$DC=4x$

$BD^1=2\sqrt{26}x$

Calculate $AA^1$ .

Video Solution

Answer

$x\sqrt{39}$