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.

(1 point)
Responses

the surface area of the cube
the surface area of the cube

the area of one of the cube’s faces
the area of one of the cube’s faces

the diagonal of one of the cube’s faces
the diagonal of one of the cube’s faces

the volume of the cube
the volume of the cube

Answer 1

To determine which measurement will require the use of irrational numbers, let's examine each option:

1. **Surface area of the cube**: The surface area \( S \) of a cube with edge length \( a \) is calculated as \( S = 6a^2 \). Since \( a \) is rational, \( a^2 \) is also rational, and thus the surface area is rational.

2. **Area of one of the cube’s faces**: The area \( A \) of one face of the cube is calculated as \( A = a^2 \). Again, since \( a \) is rational, \( a^2 \) is rational, so the area of one face is also rational.

3. **Diagonal of one of the cube’s faces**: The diagonal \( d \) of one face of the cube (which is a square) can be calculated using the Pythagorean theorem: \( d = a\sqrt{2} \). Since \( \sqrt{2} \) is irrational and \( a \) is rational, the product \( a\sqrt{2} \) is irrational. Therefore, the diagonal of one of the cube's faces requires the use of irrational numbers.

4. **Volume of the cube**: The volume \( V \) of the cube is calculated as \( V = a^3 \). Since \( a \) is rational, \( a^3 \) is also rational, so the volume is rational.

Based on the analysis, the measurement that requires the use of irrational numbers is:

**the diagonal of one of the cube’s faces.**