The Application of the Pythagorean Theorem to an Orthohedron or Cuboid

🏆Practice use of the pythagorean theorem in the 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.

A1- Uses of the Pythagorean theorem in an orthohedron

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Test yourself on use of the pythagorean theorem in the orthohedron!

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Look at the orthohedron in the figure below.

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

BBBCCCGGGFFFAAADDDHHHEEE

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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 6 , 8 8 and 10 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 6 and 8 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 X=10

That is, the diagonal measures 10 10 .

Uses of the Pythagorean theorem in an orthohedron


Examples and exercises with solutions on the application of the Pythagorean theorem in a rectangular prism or cuboid

Exercise #1

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

Calculate the diagonal of the rectangular prism.

777101010444AAABBBCCCDDDAAA111BBB111CCC111DDD111

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 in the figure below.

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

BBBCCCGGGFFFAAADDDHHHEEE

Video Solution

Answer

HBE HBE

Exercise #3

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.

777555AAABBBCCCDDDAAA111BBB111CCC111DDD111

Video Solution

Answer

Not enough data

Exercise #4

Look at the orthohedron in the figure below.

DCC1D1 DCC^1D^1 is a square.

How long is the dotted line?

121212555DDDAAABBBCCCD1D1D1A1A1A1B1B1B1C1C1C1

Video Solution

Answer

13 13

Exercise #5

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

444777

Video Solution

Answer

65 \sqrt{65}

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