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

Question Types:
Using the Pythagorean Theorem in Cuboids: Determining the size of the diagonal using Pythagoras' theoremUsing the Pythagorean Theorem in Cuboids: Using Pythagoras' theorem to calculate the height of a boxUsing the Pythagorean Theorem in Cuboids: Using Pythagoras' theorem- with parameters to calculate the lines of the box

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

Suggested Topics to Practice in Advance

  1. The Pythagorean Theorem

Practice Using the Pythagorean Theorem in Cuboids

Question 1

Shown below is the rectangular prism \( ABCDA^1B^1C^1D^1 \).

Calculate the diagonal of the rectangular prism.

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Question 2

Look at the orthohedron below.

\( D^1C^1=10 \)

\( AA^1=12 \)

Calculate \( A^1B \).

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Question 3

Look at the orthohedron in the figure below.

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

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Question 4

\( ABCDA^1B^1C^1D^1 \) is a rectangular prism.

\( AB=7 \)
\( AA^1=5 \)

Calculate the diagonal of the rectangular prism.

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Question 5

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

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Examples with solutions for Using the Pythagorean Theorem in Cuboids

Exercise #1

Shown below is the rectangular prism ABCDA1B1C1D1 ABCDA^1B^1C^1D^1 .

Calculate the diagonal of the rectangular prism.

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Video Solution

Step-by-Step Solution

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

D1C12+CC12=D1C2 D_1C_1^2+CC_1^2=D_1C^2

Let's insert the known data:

102+42=D1C2 10^2+4^2=D_1C^2

116=D1C2 116=D_1C^2

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

D1C2+CB2=BD12 D_1C^2+CB^2=BD_1^2

Let's insert the known data:

116+72=BD12 116+7^2=BD_1^2

116+49=BD12 116+49=BD_1^2

165=BD12 165=BD_1^2

Let's find the root:

165=BD1 \sqrt{165}=BD_1

Answer

165 \sqrt{165}

Exercise #2

Look at the orthohedron below.

D1C1=10 D^1C^1=10

AA1=12 AA^1=12

Calculate A1B A^1B .

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Video Solution

Step-by-Step Solution

From the given data, we can conclude that:

D1C1=A1B1=AB=10 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:

AA12+AB2=A1B2 AA_1^2+AB^2=A_1B^2

Let's substitute the known values:

122+102=A1B2 12^2+10^2=A_1B^2

A1B2=144+100=244 A_1B^2=144+100=244

Let's take the square root:

A1B=244 A_1B=\sqrt{244}

A1B=4×61=4×61 A_1B=\sqrt{4\times61}=\sqrt{4}\times\sqrt{61}

A1B=261 A_1B=2\sqrt{61}

Answer

261 2\sqrt{61}

Exercise #3

Look at the orthohedron in the figure below.

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

BBBCCCGGGFFFAAADDDHHHEEE

Video Solution

Answer

HBE HBE

Exercise #4

ABCDA1B1C1D1 ABCDA^1B^1C^1D^1 is a rectangular prism.

AB=7 AB=7
AA1=5 AA^1=5

Calculate the diagonal of the rectangular prism.

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Video Solution

Answer

Not enough data

Exercise #5

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

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Video Solution

Answer

65 \sqrt{65}

Question 1

Look at the orthohedron in the figure below.

\( DCC^1D^1 \) is a square.

How long is the dotted line?

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Question 2

Look at the box in the drawing and calculate the indicated diagonal.777888

Question 3

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

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Question 4

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

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Question 5

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

AAABBBCCCDDDEEEFFFGGGHHH

Exercise #6

Look at the orthohedron in the figure below.

DCC1D1 DCC^1D^1 is a square.

How long is the dotted line?

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Video Solution

Answer

13 13

Exercise #7

Look at the box in the drawing and calculate the indicated diagonal.777888

Video Solution

Answer

113 \sqrt{113}

Exercise #8

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

3X3X3XXXX

Video Solution

Answer

x10 x\sqrt{10}

Exercise #9

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

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Video Solution

Answer

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

Exercise #10

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

AAABBBCCCDDDEEEFFFGGGHHH

Video Solution

Answer

CD2+FG2+EA2 \sqrt{CD^2+FG^2+EA^2}

Question 1

Look at the orthohedron in the figure.

\( A^1B=12 \)

Canculate \( CC^1 \).

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Question 2

Look at the rectangular prism below.

\( ABCDA^1B^1C^1D^1 \)

\( BC^1=12 \)

\( D^1D=10 \)

\( D^1C=7 \)

Calculate AD.

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Question 3

Look at the rectangular prism in the figure below.

Calculate AD.

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Question 4

\( ABCDA^1B^1C^1D^1 \) is a rectangular prism.


\( D^1C^1=14 \)

\( BD=20 \)

Calculate the height of the rectangular prism.

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Question 5

Look at the orthohedron below.

\( DC=DD_1 \)

What is the length of the orthohedron's diagonal?

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Exercise #11

Look at the orthohedron in the figure.

A1B=12 A^1B=12

Canculate CC1 CC^1 .

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Video Solution

Answer

211 2\sqrt{11}

Exercise #12

Look at the rectangular prism below.

ABCDA1B1C1D1 ABCDA^1B^1C^1D^1

BC1=12 BC^1=12

D1D=10 D^1D=10

D1C=7 D^1C=7

Calculate AD.

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Video Solution

Answer

211 2\sqrt{11} cm

Exercise #13

Look at the rectangular prism in the figure below.

Calculate AD.

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Video Solution

Answer

222 2\sqrt{22} cm

Exercise #14

ABCDA1B1C1D1 ABCDA^1B^1C^1D^1 is a rectangular prism.


D1C1=14 D^1C^1=14

BD=20 BD=20

Calculate the height of the rectangular prism.

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Video Solution

Answer

251 2\sqrt{51} cm

Exercise #15

Look at the orthohedron below.

DC=DD1 DC=DD_1

What is the length of the orthohedron's diagonal?

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Video Solution

Answer

264 \sqrt{264}

Topics learned in later sections

  1. Congruence of Right Triangles (using the Pythagorean Theorem)