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

Understanding Using the Pythagorean Theorem in Cuboids

Complete explanation with examples

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=c2a²+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:

  1. Use the Pythagorean theorem twice: First, find one face diagonal, then use it to calculate the space diagonal.
  2. Use the formula:
    d=l2+w2+h2d=l^2+w^2+h^2​
    whereII, ww, and hh are the length, width, and height of the cuboid, respectively.
Detailed explanation

Practice Using the Pythagorean Theorem in Cuboids

Test your knowledge with 10 quizzes

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?

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Examples with solutions for Using the Pythagorean Theorem in Cuboids

Step-by-step solutions included
Exercise #1

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

Calculate the diagonal of the rectangular prism.

777101010444AAABBBCCCDDDAAA111BBB111CCC111DDD111

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}

Video Solution
Exercise #2

Look at the orthohedron below.

D1C1=10 D^1C^1=10

AA1=12 AA^1=12

Calculate A1B A^1B .

101010121212AAABBBCCCDDDAAA111BBB111CCC111DDD111

Step-by-Step Solution

From the given data, we know 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

Then we will substitute in the known values:

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

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

Finally, we calculate 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}

Video Solution
Exercise #3

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

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Step-by-Step Solution

We will use the Pythagorean theorem to find diagonal AD1:

AA1+A1D1=AD1 AA_1+A_1D_1=AD_1

Let's input the known data:

52+72=D1A2 5^2+7^2=D_1A^2

D1A2=25+49=74 D_1A^2=25+49=74

Let's find the square root:

AD1=74 AD_1=\sqrt{74}

From the data we can see that:

AA1=DD1=5 AA_1=DD_1=5

Now let's look at triangle DD1C1 and calculate DC1 using the Pythagorean theorem:

D1D2+D1C12=C1D2 D_1D^2+D_1C_1^2=C_1D^2

Let's input the existing data:

52+42=C1D2 5^2+4^2=C_1D^2

C1D2=25+16=41 C_1D^2=25+16=41

Let's find the square root:

DC1=41 DC_1=\sqrt{41}

Now let's focus on triangle A1D1C1 and find diagonal A1C1:

A1D12+D1C12=A1C12 A_1D_1^2+D_1C_1^2=A_1C_1^2

Let's input the known data:

72+42=A1C12 7^2+4^2=A_1C_1^2

A1C12=49+16=65 A_1C_1^2=49+16=65

Let's find the square root:

A1C1=65 A_1C_1=\sqrt{65}

Now we have all 3 lengths of all possible diagonal corners in the box:

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

Answer:

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

Video Solution
Exercise #4

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

Calculate the length of the side AC.

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Step-by-Step Solution

To find the length of the diagonal AC AC on a cuboid, we will use the Pythagorean theorem twice:

  • Step 1: Calculate AB AB , the diagonal of the face of the cuboid:

We assume the cuboid's known dimensions, with width 8cm 8 \, \text{cm} and height 4cm 4 \, \text{cm} . Assume l l for one dimension along the base.

Since AB=w2+l2 AB = \sqrt{w^2 + l^2} , solving for AB means understanding both directions. As the length isn't given, we solve specifically for vertically so AC AC depends on full volume space:

  • Step 2: Calculate diagonal AC AC :

AC=AB2+height2=82+42=64+16=80cm AC = \sqrt{AB^2 + \text{height}^2} = \sqrt{8^2 + 4^2} = \sqrt{64 + 16} = \sqrt{80} \, \text{cm}

In this way, we determine the length of diagonal AC AC is 80 \sqrt{80} cm.

The correct choice corresponding to this calculation is Choice 3: 80 \sqrt{80} cm.

Answer:

80 \sqrt{80} cm

Video Solution
Exercise #5

Look at the orthohedron in the figure below.

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

BBBCCCGGGFFFAAADDDHHHEEE

Step-by-Step Solution

Answer:

HBE HBE

Video Solution

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