Calculate the Face Diagonal of a 4cm Cube: Finding the Corner-to-Corner Distance

Face Diagonal with Pythagorean Theorem

Below is a cube with edges equal to 4 cm.

What is the length of the diagonal of the cube's face indicated in the figure?

444

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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
00:10 We'll substitute appropriate values and solve for the diagonal
00:32 We'll break down the square root of 32 into square root of 16 times square root of 2
00:38 We'll calculate the root
00:44 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Below is a cube with edges equal to 4 cm.

What is the length of the diagonal of the cube's face indicated in the figure?

444

2

Step-by-step solution

To solve for the diagonal of a face of the cube, follow these steps:

  • Identify the side length of the square face: Each side of the face of the cube is 4 cm.
  • Recognize that the diagonal forms a right triangle with two sides of the square face.
  • Apply the Pythagorean theorem: The diagonal d d is found using d=a2+b2 d = \sqrt{a^2 + b^2} where a a and b b are the sides of the square.
  • Substitute the side lengths into the formula: Since both a a and b b are 4 cm, the formula becomes d=42+42=16+16=32 d = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} .
  • Simplify the square root: 32=16×2=16×2=4×2 \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4 \times \sqrt{2} .

Therefore, the length of the diagonal of the cube's face is 4×2 4 \times \sqrt{2} cm.

3

Final Answer

4×2 4\times\sqrt{2} cm

Key Points to Remember

Essential concepts to master this topic
  • Rule: Face diagonal uses Pythagorean theorem with two equal sides
  • Technique: For 4cm cube: d=42+42=32=42 d = \sqrt{4^2 + 4^2} = \sqrt{32} = 4\sqrt{2}
  • Check: Verify (42)2=32=16+16 (4\sqrt{2})^2 = 32 = 16 + 16 ✓

Common Mistakes

Avoid these frequent errors
  • Using space diagonal formula instead of face diagonal
    Don't use a2+b2+c2 \sqrt{a^2 + b^2 + c^2} for face diagonal = wrong 3D calculation! This gives the cube's space diagonal, not the 2D face diagonal. Always use a2+b2 \sqrt{a^2 + b^2} for face diagonal on a square.

Practice Quiz

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A cube has a total of 14 edges.

FAQ

Everything you need to know about this question

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

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A face diagonal goes corner-to-corner on a flat square face (2D), while a space diagonal goes through the cube's interior from one corner to the opposite corner (3D).

Why do I get 42 4\sqrt{2} instead of a regular number?

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The diagonal of any square with side s is always s2 s\sqrt{2} . This creates an irrational number that can't be simplified to a whole number or simple fraction.

Can I just measure the diagonal instead of calculating?

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While measuring works practically, calculating gives the exact answer. Measuring might give you approximately 5.66 cm, but the exact answer is 42 4\sqrt{2} cm.

Do all cube faces have the same diagonal length?

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Yes! Since all faces of a cube are identical squares with the same side length, every face diagonal has the same length of 42 4\sqrt{2} cm.

How do I simplify 32 \sqrt{32} ?

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Factor out perfect squares: 32=16×2=16×2=42 \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} . Always look for the largest perfect square factor!

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