Question

A cube has an edge length, a, that is a rational number of centimeters. Identify which measurement will require the use of irrational numbers.

• the volume of the cube
• the surface area of the cube
O the area of one of the cube's faces
O the diagonal of one of the cube's faces

Answer 1

To assess which measurement will require the use of irrational numbers based on a cube with an edge length \( a \) that is a rational number:

1. **Volume of the Cube**: The volume \( V \) of a cube is given by the formula:
\[
V = a^3
\]
Since \( a \) is rational, \( a^3 \) will also be rational.

2. **Surface Area of the Cube**: The surface area \( SA \) of a cube is given by the formula:
\[
SA = 6a^2
\]
As \( a \) is rational, \( 6a^2 \) will also be rational.

3. **Area of One of the Cube's Faces**: The area \( A \) of one face of the cube is given by the formula:
\[
A = a^2
\]
Since \( a \) is rational, \( a^2 \) will also be rational.

4. **Diagonal of One of the Cube's Faces**: The diagonal \( d \) of a face of the cube can be calculated using the Pythagorean theorem:
\[
d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2}
\]
Here, \( \sqrt{2} \) is an irrational number. Therefore, since \( a \) is rational, \( a\sqrt{2} \) is irrational.

Based on this analysis, the measurement that will require the use of irrational numbers is:
- **The diagonal of one of the cube's faces**.