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 volume of the cube
the volume of the cube

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

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

the surface area of the cube

Answer 1

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

1. **The volume of the cube** is calculated using the formula \( V = a^3 \). Since \( a \) is a rational number, \( a^3 \) will also be a rational number.

2. **The area of one of the cube’s faces** is calculated using the formula \( A = a^2 \). Again, since \( a \) is a rational number, \( a^2 \) will also be a rational number.

3. **The surface area of the cube** is calculated using the formula \( SA = 6a^2 \). Since \( a^2 \) is rational, \( 6a^2 \) will also be rational.

4. **The diagonal of one of the cube’s faces** can be found using the formula \( d = a\sqrt{2} \). Here, \( \sqrt{2} \) is an irrational number. Therefore, \( a\sqrt{2} \) will be irrational, as a rational number multiplied by an irrational number results in an irrational number.

Based on this analysis, the measurement that will require the use of irrational numbers is:

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