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

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

the volume of the cube
the volume of the cube

the diagonal of one of the cube’s faces

Answer 1

To determine which measurement will require the use of irrational numbers, let’s analyze each option given that \( a \) is a rational number:

1. **The area of one of the cube’s faces**:
- The area of one face of the cube is given by \( a^2 \), which is rational since the square of a rational number is also rational.

2. **The surface area of the cube**:
- The surface area of the cube is given by \( 6a^2 \), which is also rational for the same reason as above.

3. **The volume of the cube**:
- The volume of the cube is given by \( a^3 \), which is again rational since the cube of a rational number is rational.

4. **The diagonal of one of the cube’s faces**:
- The diagonal \( d \) of one of the cube's faces can be calculated using the Pythagorean theorem:
\[
d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2}
\]
- Since \( \sqrt{2} \) is an irrational number, \( a\sqrt{2} \) will be irrational unless \( a = 0 \).

Therefore, the measurement that will require the use of irrational numbers is **the diagonal of one of the cube’s faces**.