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

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

Calculate the diagonal of the rectangular prism.

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:

$\sqrt{165}=BD_1$

$\sqrt{165}$

Look at the orthohedron in the figure below.

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

$HBE$

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

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

Calculate the diagonal of the rectangular prism.

Not enough data

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

$\sqrt{65}$

Look at the orthohedron in the figure below.

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

How long is the dotted line?

$13$

Look at the orthohedron below.

$D^1C^1=10$

$AA^1=12$

Calculate $A^1B$.

$2\sqrt{61}$

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

$\sqrt{113}$

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

$x\sqrt{10}$

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

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

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

$\sqrt{CD^2+FG^2+EA^2}$

Look at the orthohedron in the figure.

$A^1B=12$

Canculate $CC^1$.

$2\sqrt{11}$

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

Calculate the length of the side AC.

$\sqrt{80}$ cm

Look at the rectangular prism below.

$ABCDA^1B^1C^1D^1$

$BC^1=12$

$D^1D=10$

$D^1C=7$

Calculate AD.

$2\sqrt{11}$ cm

Look at the rectangular prism in the figure below.

Calculate AD.

$2\sqrt{22}$ cm

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


$D^1C^1=14$

$BD=20$

Calculate the height of the rectangular prism.

$2\sqrt{51}$ cm