Use of the Pythagorean Theorem in the orthohedron - Examples, Exercises and Solutions

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

Suggested Topics to Practice in Advance

  1. The Pythagorean Theorem

Practice Use of the Pythagorean Theorem in the orthohedron

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 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 and calculate the length of the dotted line.

444777

Video Solution

Answer

65 \sqrt{65}

Exercise #5

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 #1

Look at the orthohedron below.

D1C1=10 D^1C^1=10

AA1=12 AA^1=12

Calculate A1B A^1B .

101010121212AAABBBCCCDDDAAA111BBB111CCC111DDD111

Video Solution

Answer

261 2\sqrt{61}

Exercise #2

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

Video Solution

Answer

113 \sqrt{113}

Exercise #3

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

3X3X3XXXX

Video Solution

Answer

x10 x\sqrt{10}

Exercise #4

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

444777555

Video Solution

Answer

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

Exercise #5

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}

Exercise #1

The side BC in the rectangular prism below is 8 cm long.

Side BD is 4 cm long.
Side AD is 5 cm long.

Calculate the volume of the cube.

444888555DDDBBBCCCAAA

Video Solution

Answer

96 cm³

Exercise #2

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?

999333101010AAABBB

Video Solution

Answer

You can, 3619 36\cdot\sqrt{19} cm³

Exercise #3

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

Calculate the length of the side AC.

888444DDDBBBAAACCC

Video Solution

Answer

80 \sqrt{80} cm

Exercise #4

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.

101010121212777AAABBBCCCDDDAAA111BBB111CCC111DDD111

Video Solution

Answer

211 2\sqrt{11} cm

Exercise #5

Look at the rectangular prism in the figure below.

Calculate AD.

999131313AAABBBCCCDDDAAA111BBB111CCC111DDD111

Video Solution

Answer

222 2\sqrt{22} cm

Topics learned in later sections

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