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Advanced Metric Relations

Cutting Across the Face

You already know that a cube is made of six identical square faces. A face diagonal is a line that stretches from one corner of a square face to the opposite corner. Every face on a cube has two of these diagonals.

To find the length of this diagonal, we can use the Pythagorean theorem on the right triangle formed by two adjacent sides of the square and the diagonal itself. If the side length of the cube is ss, then two sides of this triangle are ss. The diagonal is the hypotenuse.

s2+s2=df22s2=df2df=2s2df=s2s^2 + s^2 = d_f^2 \\ 2s^2 = d_f^2 \\ d_f = \sqrt{2s^2} \\ d_f = s\sqrt{2}

Through the Cube's Core

Now, let's consider a longer diagonal, one that passes through the interior of the cube. This is called a space diagonal or body diagonal. It connects two opposite vertices of the cube that do not share a face.

To find the length of the space diagonal, we can apply the Pythagorean theorem again. This time, we form a right triangle inside the cube. The legs of this triangle are a side edge (ss) and a face diagonal (s2s\sqrt{2}). The space diagonal (dsd_s) is the hypotenuse.

s2+(s2)2=ds2s2+2s2=ds23s2=ds2ds=3s2ds=s3s^2 + (s\sqrt{2})^2 = d_s^2 \\ s^2 + 2s^2 = d_s^2 \\ 3s^2 = d_s^2 \\ d_s = \sqrt{3s^2} \\ d_s = s\sqrt{3}

This is the Pythagorean theorem in three dimensions. For a rectangular box with length ll, width ww, and height hh, the space diagonal is l2+w2+h2\sqrt{l^2 + w^2 + h^2}. Since a cube has l=w=h=sl=w=h=s, this simplifies to s2+s2+s2=3s2=s3\sqrt{s^2 + s^2 + s^2} = \sqrt{3s^2} = s\sqrt{3}.

Fixed Ratios and Applications

For any cube, the lengths of its side, face diagonal, and space diagonal are always in a fixed ratio to each other. This is a powerful relationship, because if you know any one of these three lengths, you can calculate the other two, as well as the cube's volume (V=s3V=s^3) and surface area (SA=6s2SA=6s^2).

MeasurementFormulaRatio to side (s)
Side lengthss11
Face diagonaldf=s2d_f = s\sqrt{2}2≈1.414\sqrt{2} \approx 1.414
Space diagonalds=s3d_s = s\sqrt{3}3≈1.732\sqrt{3} \approx 1.732

Imagine you're building a cubic frame for a sculpture and need to add a reinforcing rod that runs diagonally through the center. The frame has sides of 2 meters.

To find the length of the rod, you would calculate the space diagonal.

ds=s3d_s = s\sqrt{3} ds=23≈3.46d_s = 2\sqrt{3} \approx 3.46 meters

What if you only knew that the longest rod that could fit inside the box was 5 meters long? You could find the side length by rearranging the formula.

5=s35 = s\sqrt{3} s=53≈2.89s = \frac{5}{\sqrt{3}} \approx 2.89 meters

From this single interior measurement, you can determine the cube's exact dimensions.

Ready to test your understanding?

Quiz Questions 1/5

A cube has a side length, ss. Which formula correctly calculates the length of its face diagonal, dfd_f?

Quiz Questions 2/5

What is the term for a diagonal that passes through the interior of the cube, connecting two opposite vertices?

Understanding these internal diagonals moves beyond the surface of a cube, giving you the tools to analyze its complete three-dimensional structure.