Space Diagonal of a Cube Formula
The space diagonal of a cube is the longest diagonal that stretches from one corner of the cube to the opposite corner, passing through its interior. This diagonal can be calculated if the side length of the cube is known, as it forms the hypotenuse of a three-dimensional right triangle within the cube.
In this guide, we will go through the formula for calculating the space diagonal of a cube, along with a detailed explanation and examples.
Formula for the Space Diagonal of a Cube
Using Side Length:
If the side length \( a \) of the cube is known,
The space diagonal \( d_s \) can be calculated with the formula:
\( d_s = a\sqrt{3} \)
In this formula:
- \( a \) is the length of one side of the cube
- \( \sqrt{3} \) is the square root of 3, approximately equal to 1.732
Detailed Explanation of the Formula
Understanding the Formula
The formula \( d_s = a\sqrt{3} \) calculates the space diagonal of a cube by multiplying the side length by \( \sqrt{3} \). This approach works because the space diagonal forms the hypotenuse of a right triangle with three edges of the cube. Using the Pythagorean theorem in three dimensions, we get \( d_s = \sqrt{a^2 + a^2 + a^2} = a\sqrt{3} \).
Example 1: Calculating Space Diagonal with Given Side Length
Problem: Find the space diagonal of a cube with a side length of \( a = 4 \, \text{cm} \).
Solution:
- Write down the formula: \( d_s = a\sqrt{3} \).
- Substitute \( a = 4 \): \( d_s = 4 \times \sqrt{3} \).
- Approximate \( \sqrt{3} \approx 1.732 \): \( d_s \approx 4 \times 1.732 = 6.928 \, \text{cm} \).
The space diagonal of the cube is approximately \( 6.928 \, \text{cm} \).
Example 2: Finding Side Length from Space Diagonal
Problem: A cube has a space diagonal of \( d_s = 9 \, \text{cm} \). Find the side length.
Solution:
- Start with the space diagonal formula: \( d_s = a\sqrt{3} \).
- Rearrange to solve for \( a \): \( a = \dfrac{d_s}{\sqrt{3}} \).
- Substitute \( d_s = 9 \): \( a = \dfrac{9}{\sqrt{3}} \).
- Approximate \( \sqrt{3} \approx 1.732 \): \( a \approx \dfrac{9}{1.732} \approx 5.2 \, \text{cm} \).
The side length of the cube is approximately \( 5.2 \, \text{cm} \).
These examples demonstrate how to calculate the space diagonal of a cube or find an unknown side length based on the space diagonal formula \( d_s = a\sqrt{3} \).