Using the Pythagorean Theorem in Cuboids: Determining the size of the diagonal using Pythagoras' theorem

Using Pythagoras' theorem to calculate the height of a box Using Pythagoras' theorem- with parameters to calculate the lines of the box

Question 1

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

Calculate the diagonal of the rectangular prism.

Question 2

Look at the orthohedron below.

$$ D^1C^1=10 $$

$$ AA^1=12 $$

Calculate $$ A^1B $$ .

Question 3

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

Question 4

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

$$ AB=7 $$
$$ AA^1=5 $$

Calculate the diagonal of the rectangular prism.

Question 5

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

Examples with solutions for Using the Pythagorean Theorem in Cuboids: Determining the size of the diagonal using Pythagoras' theorem

Exercise #1

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

Calculate the diagonal of the rectangular prism.

Video Solution

Step-by-Step Solution

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

$D_1C_1^2+CC_1^2=D_1C^2$

Let's insert the known data:

$10^2+4^2=D_1C^2$

$116=D_1C^2$

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

$D_1C^2+CB^2=BD_1^2$

Let's insert the known data:

$116+7^2=BD_1^2$

$116+49=BD_1^2$

$165=BD_1^2$

Let's find the root:

$\sqrt{165}=BD_1$

Answer

$\sqrt{165}$

Exercise #2

Look at the orthohedron below.

$D^1C^1=10$

$AA^1=12$

Calculate $A^1B$ .

Video Solution

Step-by-Step Solution

From the given data, we know that:

$D_1C_1=A_1B_1=AB=10$

Let's draw a diagonal between A1 and B and focus on triangle AA1B.

We'll calculate A1B using the Pythagorean theorem:

$AA_1^2+AB^2=A_1B^2$

Then we will substitute in the known values:

$12^2+10^2=A_1B^2$

$A_1B^2=144+100=244$

Finally, we calculate square root:

$A_1B=\sqrt{244}$

$A_1B=\sqrt{4\times61}=\sqrt{4}\times\sqrt{61}$

$A_1B=2\sqrt{61}$

Answer

$2\sqrt{61}$

Exercise #3

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

Video Solution

Step-by-Step Solution

We will use the Pythagorean theorem to find diagonal AD1:

$AA_1+A_1D_1=AD_1$

Let's input the known data:

$5^2+7^2=D_1A^2$

$D_1A^2=25+49=74$

Let's find the square root:

$AD_1=\sqrt{74}$

From the data we can see that:

$AA_1=DD_1=5$

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

$D_1D^2+D_1C_1^2=C_1D^2$

Let's input the existing data:

$5^2+4^2=C_1D^2$

$C_1D^2=25+16=41$

Let's find the square root:

$DC_1=\sqrt{41}$

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

$A_1D_1^2+D_1C_1^2=A_1C_1^2$

Let's input the known data:

$7^2+4^2=A_1C_1^2$

$A_1C_1^2=49+16=65$

Let's find the square root:

$A_1C_1=\sqrt{65}$

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

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

Answer

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

Exercise #4

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

$AB=7$
$AA^1=5$

Calculate the diagonal of the rectangular prism.

Video Solution

Answer

Not enough data

Exercise #5

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

Video Solution

Answer

$\sqrt{65}$

Question 1

Look at the orthohedron in the figure below.

$$ DCC^1D^1 $$ is a square.

How long is the dotted line?

Question 2

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

Question 3

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

Question 4

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

Answer

$\sqrt{264}$

Question 1

The box $$ ABCDA^1B^1C^1D^1 $$ is shown below.

$$ A^1B^1=14 $$

$$ CC^1=8 $$

$$ A^1D^1=9 $$

Calculate the diagonal of the box.

Question 2

A cube has a side length of 5 cm.

Calculate the diagonal of the cube.

Question 3

Look at the rectangular prism in the figure and express the length of its diagonal terms of a and b.

Question 4

Look at the orthohedron in the figure and calculate $$ B^1D $$ .

Question 5

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?

Answer

$\sqrt{41}$