Use the image to answer the question.

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 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

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

the volume of the cube

Answer 1

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

This is because the diagonal \(d\) of a square face of the cube can be calculated using the Pythagorean theorem:

\[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]

Since \(a\) is a rational number, \(a\sqrt{2}\) will be irrational because \(\sqrt{2}\) is an irrational number. Therefore, the correct response is the diagonal of one of the cube’s faces.