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

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?

Video Solution

Solution Steps

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 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 $d$ is found using $d = \sqrt{a^2 + b^2}$ where $a$ and $b$ are the sides of the square.
Substitute the side lengths into the formula: Since both $a$ and $b$ are 4 cm, the formula becomes $d = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32}$ .
Simplify the square root: $\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4 \times \sqrt{2}$ .

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