Calculate the Face Diagonal of a 5cm Cube Using the Pythagorean Theorem

Q: Why can't the face diagonal just be 5 cm like the edge?

The diagonal goes corner to corner across the face, which is longer than going along an edge. Think of it as the hypotenuse of a right triangle with legs of 5 cm each.

Q: Where does the √2 come from in the formula?

It comes from the Pythagorean theorem! For a square with side s: $ d^2 = s^2 + s^2 = 2s^2 $, so $ d = s\sqrt{2} $.

Q: Do I need to memorize the formula d = s√2?

It helps, but you can always derive it! Just draw the diagonal in the square face and use $ a^2 + b^2 = c^2 $ with both legs equal to the edge length.

Q: What's the difference between face diagonal and space diagonal?

A face diagonal goes across one square face (like this problem). A space diagonal goes from one corner of the cube to the opposite corner through the interior.

Q: Should I leave my answer as 5√2 or calculate the decimal?

Keep it as $ 5\sqrt{2} $ cm unless asked for a decimal approximation. The exact form is more precise and shows your mathematical reasoning clearly.

Face Diagonal Calculations with Square Applications

Shown below is a cube with edges equaling 5 cm.

What is the length of the diagonal on the cube's face?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the diagonal of the cube face

00:03 We'll use the Pythagorean theorem in triangle ABD

00:13 We'll substitute appropriate values and solve to find the diagonal

00:43 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Shown below is a cube with edges equaling 5 cm.

What is the length of the diagonal on the cube's face?

Step-by-step solution

To solve this problem, we need to determine the length of the diagonal of one face of a cube with edge length 5 cm.

Step 1: Recognize that one face of the cube is a square with side length 5 cm.
Step 2: Apply the Pythagorean theorem formula for the diagonal $d$ of a square, $d = s\sqrt{2}$ , where $s$ is the side length.
Step 3: Substitute $s = 5$ cm into the formula to compute the diagonal.

Now, let's perform the calculations:
The diagonal $d$ of the square (face of the cube) is given by:

$d = 5\sqrt{2}$ .

Therefore, the length of the diagonal on the cube's face is $5\sqrt{2}$ cm.

Final Answer

$5\times\sqrt{2}$ cm

Key Points to Remember

Essential concepts to master this topic

Rule: Face diagonal equals side length times square root of 2
Technique: Apply $d = s\sqrt{2}$ where s = 5 gives $5\sqrt{2}$ cm
Check: Pythagorean theorem: $5^2 + 5^2 = (5\sqrt{2})^2 = 50$ ✓

Common Mistakes

Avoid these frequent errors

Using edge length as diagonal length
Don't think the face diagonal equals 5 cm = wrong by factor of √2! The diagonal is always longer than the edge. Always use the diagonal formula $d = s\sqrt{2}$ for squares.

Practice Quiz

Test your knowledge with interactive questions

A cube has a total of 14 edges.

FAQ

Everything you need to know about this question

Why can't the face diagonal just be 5 cm like the edge?

The diagonal goes corner to corner across the face, which is longer than going along an edge. Think of it as the hypotenuse of a right triangle with legs of 5 cm each.

Where does the √2 come from in the formula?

It comes from the Pythagorean theorem! For a square with side s: $d^2 = s^2 + s^2 = 2s^2$ , so $d = s\sqrt{2}$ .

Do I need to memorize the formula d = s√2?

It helps, but you can always derive it! Just draw the diagonal in the square face and use $a^2 + b^2 = c^2$ with both legs equal to the edge length.

What's the difference between face diagonal and space diagonal?

A face diagonal goes across one square face (like this problem). A space diagonal goes from one corner of the cube to the opposite corner through the interior.

Should I leave my answer as 5√2 or calculate the decimal?

Keep it as $5\sqrt{2}$ cm unless asked for a decimal approximation. The exact form is more precise and shows your mathematical reasoning clearly.

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