Question

Use the image to answer the question.

The length, width, and height of a cube are all labeled a.

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 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 surface area of the cube
the surface area of the cube

Answer 1

To determine which measurement will require the use of irrational numbers in the context of a cube with edge length \( a \) (where \( a \) is a rational number), we will analyze each option:

1. **The area of one of the cube’s faces**: The area of one face of a cube is calculated as \( a \times a = a^2 \). Since \( a^2 \) is a rational number (the square of a rational number is rational), this measurement does not involve irrational numbers.

2. **The volume of the cube**: The volume of the cube is calculated as \( a \times a \times a = a^3 \). Since \( a^3 \) is also a rational number (the cube of a rational number is rational), this measurement does not involve irrational numbers either.

3. **The diagonal of one of the cube’s faces**: The diagonal of a square face 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, \( d \) is a product of a rational number \( a \) and an irrational number \( \sqrt{2} \), making \( d \) irrational.

4. **The surface area of the cube**: The surface area of the cube is calculated as \( 6 \times a^2 \). Since \( a^2 \) is rational, \( 6a^2 \) is also a rational number. Hence, this measurement does not involve irrational numbers.

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

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