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

Q: How many different face diagonals are there in a rectangular prism?

A rectangular prism has 3 types of face diagonals because it has 3 different rectangular faces. Each type appears on multiple faces, but the lengths are what matter for this problem.

Q: Which dimensions do I use for each diagonal?

Look at each rectangular face separately:5×7 face: $ \sqrt{5^2 + 7^2} = \sqrt{74} $4×5 face: $ \sqrt{4^2 + 5^2} = \sqrt{41} $4×7 face: $ \sqrt{4^2 + 7^2} = \sqrt{65} $

Q: Can I simplify these square roots further?

Let's check each one:$ \sqrt{74} $ cannot be simplified (74 = 2 × 37)$ \sqrt{41} $ cannot be simplified (41 is prime)$ \sqrt{65} $ cannot be simplified (65 = 5 × 13)So all three answers stay as exact values with square roots.

Q: How is this different from finding the space diagonal?

A space diagonal goes through the inside of the prism from one corner to the opposite corner, using all three dimensions. Face diagonals only go across the surface of rectangular faces using just two dimensions.

Face Diagonals with Pythagorean Theorem

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

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

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 written solution

Follow each step carefully to understand the complete solution

Understand the problem

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

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:

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

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Concept: Face diagonals connect non-adjacent vertices on rectangular faces
Formula: Use $d = \sqrt{a^2 + b^2}$ for rectangle with sides a, b
Check: Verify each diagonal by squaring result: $(\sqrt{74})^2 = 5^2 + 7^2 = 74$ ✓

Common Mistakes

Avoid these frequent errors

Confusing face diagonals with space diagonals
Don't try to use all three dimensions at once = wrong calculation! Face diagonals only use two dimensions from the rectangle's sides. Always identify which face you're working with first, then use only those two dimensions.

Practice Quiz

Test your knowledge with interactive questions

Consider a right-angled triangle, AB = 8 cm and AC = 6 cm.
Calculate the length of side BC.

FAQ

Everything you need to know about this question

How many different face diagonals are there in a rectangular prism?

A rectangular prism has 3 types of face diagonals because it has 3 different rectangular faces. Each type appears on multiple faces, but the lengths are what matter for this problem.

Why do we get three different diagonal lengths?

Because the prism has dimensions 4×5×7, we get three different rectangular faces: 4×5, 5×7, and 4×7. Each rectangle produces a different diagonal length using the Pythagorean theorem.

Which dimensions do I use for each diagonal?

Look at each rectangular face separately:

5×7 face: $\sqrt{5^2 + 7^2} = \sqrt{74}$
4×5 face: $\sqrt{4^2 + 5^2} = \sqrt{41}$
4×7 face: $\sqrt{4^2 + 7^2} = \sqrt{65}$

Can I simplify these square roots further?

Let's check each one:

$\sqrt{74}$ cannot be simplified (74 = 2 × 37)
$\sqrt{41}$ cannot be simplified (41 is prime)
$\sqrt{65}$ cannot be simplified (65 = 5 × 13)

So all three answers stay as exact values with square roots.

How is this different from finding the space diagonal?

A space diagonal goes through the inside of the prism from one corner to the opposite corner, using all three dimensions. Face diagonals only go across the surface of rectangular faces using just two dimensions.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Pythagorean Theorem questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

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

Face Diagonals with Pythagorean Theorem

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How many different face diagonals are there in a rectangular prism?

Why do we get three different diagonal lengths?

Which dimensions do I use for each diagonal?

Can I simplify these square roots further?

How is this different from finding the space diagonal?

🌟 Unlock Your Math Potential

More Questions

Using the Pythagorean Theorem in Cuboids

Calculate A¹B in an Orthohedron: Given D¹C¹=10 and AA¹=12

Calculate the Diagonal of a 7×10×4 Rectangular Prism: Space Geometry Problem