Calculate Face Diagonals: Rectangular Prism with Dimensions 4×5×7

Question

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

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Video Solution

Solution Steps

00:00 Find all possible diagonal sides
00:03 The box's face is a rectangle
00:13 Use the Pythagorean theorem in triangle AA1D1 to find AD1
00:20 Substitute appropriate values according to the given data and solve for AD1
00:33 This is one possible diagonal length
00:42 The box's face is a rectangle, therefore opposite sides are equal
00:50 Use the Pythagorean theorem in triangle DD1C1 to find DC1
00:57 Substitute appropriate values according to the given data and solve for DC1
01:09 This is a second possible diagonal length
01:21 Use the Pythagorean theorem in triangle A1D1C1 to find A1C1
01:31 Substitute appropriate values according to the given data and solve for A1C1
01:46 This is a third possible diagonal length
01:51 And this is the solution to the question

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}