Pythagorean Theorem in Cuboids Practice Problems

Master 3D applications of the Pythagorean theorem with cuboid diagonal problems. Practice finding face diagonals and space diagonals in rectangular prisms.

📚What You'll Master in This Practice Session
  • Calculate face diagonal lengths using cuboid dimensions
  • Find space diagonals through interior of rectangular prisms
  • Apply two-step Pythagorean theorem method for 3D problems
  • Use the direct space diagonal formula d² = l² + w² + h²
  • Solve real-world problems involving box and container measurements
  • Master transition from 2D to 3D Pythagorean applications

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+h2​d=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

A rectangular prism has a height twice as long as its length.

Its width is 3 times greater than its height.

How many times greater is its diagonal than its length?

6X6X6XXXX2X2X2XAAABBBCCCDDDEEEFFFGGGHHH

Examples with solutions for Using the Pythagorean Theorem in Cuboids

Step-by-step solutions included
Exercise #1

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 #2

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 #3

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

444777555

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.

888444DDDBBBAAACCC

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 8 cm 8 \, \text{cm} and height 4 cm 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=80 cm 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

Frequently Asked Questions

What is the difference between face diagonal and space diagonal in a cuboid?

+
A face diagonal lies on one of the rectangular faces of the cuboid, while a space diagonal runs through the interior from one corner to the opposite corner. Face diagonals use 2D Pythagorean theorem, but space diagonals require 3D calculations using all three dimensions.

How do you find the space diagonal of a rectangular prism step by step?

+
Method 1: First find a face diagonal using two dimensions, then use that diagonal with the third dimension. Method 2: Use the direct formula d² = l² + w² + h² where l, w, h are length, width, and height.

Why can we use the Pythagorean theorem in cuboids?

+
Cuboids have right angles between all adjacent faces, creating right triangles when we draw diagonals. This makes the Pythagorean theorem applicable for calculating diagonal lengths in these 3D shapes.

What is the formula for space diagonal in a cuboid?

+
The space diagonal formula is d = √(l² + w² + h²), where d is the diagonal length, and l, w, h represent the three dimensions of the cuboid.

How many face diagonals does a cuboid have?

+
A cuboid has 12 face diagonals total: 4 diagonals on each pair of opposite faces. Each rectangular face has 2 diagonals, and there are 6 faces, but opposite faces have equal diagonal lengths.

Can you use Pythagorean theorem for any 3D shape?

+
No, the Pythagorean theorem only works with right triangles. In 3D shapes, you can only use it when the shape has perpendicular edges that form right angles, like cuboids, cubes, and rectangular prisms.

What are real-world applications of cuboid diagonal calculations?

+
Common applications include: calculating shipping box capacities, determining if long items fit diagonally in containers, designing storage spaces, measuring room diagonal distances, and solving construction problems involving rectangular structures.

Is there a shortcut to find all diagonals in a cube?

+
Yes! In a cube with side length s: face diagonals = s√2, and space diagonal = s√3. Since all edges are equal, you only need one measurement to find all diagonal lengths.

Continue Your Math Journey

Suggested Topics to Practice in Advance

Practice by Question Type