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Diagonal of A Cube Formula: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Diagonal of A Cube FormulaDiagonal of A Cube Formula: Definition and ExamplesTable of ContentsDiagonal of a Cube Formula Definition of Diagonal of a Cube

A diagonal of a cube is a line segment that connects any pair of non-adjacent vertices of the cube. There are two main types of diagonals in a cube: face diagonals and body diagonals. Face diagonals connect non-adjacent vertices on the same face of the cube, with each of the six square faces having two face diagonals, resulting in a total of 121212 face diagonals. Body diagonals, also known as space diagonals or internal diagonals, connect opposite vertices of the cube by passing through the interior of the cube, and there are 444 body diagonals in total.

For a cube with side length "aaa", the formula for the face diagonal is df=a2d_f = a\sqrt{2}df​=a2​, while the formula for the body diagonal is db=a3d_b = a\sqrt{3}db​=a3​. These formulas are derived using the Pythagorean theorem. For the face diagonal, we form a right-angled triangle with two sides of length "aaa" and the face diagonal as the hypotenuse. For the body diagonal, we form a right-angled triangle with one side of length "aaa", another side being the face diagonal (a2a\sqrt{2}a2​), and the body diagonal as the hypotenuse.

Examples of Diagonal of a Cube Formula Example 1: Finding the Face Diagonal of a Cube Problem:

Calculate the length of the face diagonal of a cube whose each side is 888 inches long.

Step-by-step solution:

Step 1, Write down what we know. The side length of the cube is a=8a = 8a=8 inches.

Step 2, Recall the formula for the face diagonal of a cube. The face diagonal =a2= a\sqrt{2}=a2​.

Step 3, Put the value of side length into the formula.

df=2×8d_f = \sqrt{2} \times 8df​=2​×8

Step 4, Calculate the value of 2\sqrt{2}2​ (approximately 1.411.411.41).

df=1.41×8d_f = 1.41 \times 8df​=1.41×8

Step 5, Multiply to find the final answer.

df=11.28d_f = 11.28df​=11.28 inches Example 2: Calculating the Body Diagonal of a Cube Problem:

Calculate the length of the body diagonal of a cube with side-length of 999 inches.

Step-by-step solution:

Step 1, Write down what we know. The side length of the cube is a=9a = 9a=9 inches.

Step 2, Recall the formula for the body diagonal of a cube. The body diagonal =a3= a\sqrt{3}=a3​.

Step 3, Put the value of side length into the formula.

db=3×9d_b = \sqrt{3} \times 9db​=3​×9

Step 4, Calculate the value of 3\sqrt{3}3​ (approximately 1.731.731.73).

db=1.73×9d_b = 1.73 \times 9db​=1.73×9

Step 5, Multiply to find the final answer.

db=15.57d_b = 15.57db​=15.57 inches Example 3: Finding Cube Dimensions from Body Diagonal Problem:

The body diagonal of a cube is 434\sqrt{3}43​ units. What will be the length side of the cube and the length of the face diagonal of the cube?

Step-by-step solution:

Step 1, Write down what we know. The body diagonal =43= 4\sqrt{3}=43​ units.

Step 2, Recall the formula for the body diagonal. The body diagonal =a3= a\sqrt{3}=a3​.

Step 3, Set up an equation with the known body diagonal.

a3=43a\sqrt{3} = 4\sqrt{3}a3​=43​

Step 4, Solve for the side length by dividing both sides by 3\sqrt{3}3​.

a=4a = 4a=4 units

Step 5, Find the face diagonal using the formula df=a2d_f = a\sqrt{2}df​=a2​.

df=2×4d_f = \sqrt{2} \times 4df​=2​×4

Step 6, Calculate the value of 2\sqrt{2}2​ (approximately 1.411.411.41).

df=1.41×4d_f = 1.41 \times 4df​=1.41×4

Step 7, Multiply to find the final answer.

df=5.64d_f = 5.64df​=5.64 units Comments(2)TTravelBugFinnNovember 6, 2025This glossary page on the diagonal of a cube formula is great! It helped my student grasp the concept easily. Clear and useful examples.

MManagerDanNovember 4, 2025This glossary page on the diagonal of a cube formula is great! It helped my students grasp the concepts easily. Thanks for the clear examples!

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