SAT Mathematics Practice Test 5

A cube has 6 faces of equal area, so if the total surface area of a cube is $6{\left(\frac{\mathit{a}}{4}\right)}^2$, then the area of one face is ${\left(\frac{\mathit{a}}{4}\right)}^2$.Likewise, the area of one face of a cube is the square of one of its sides; therefore, if the area of one face is ${\left(\frac{\mathit{a}}{4}\right)}^2$ , then the length of one side of thecube is $\frac{\mathit{a}}{4}$ Since the perimeter of one face of a cube is four times the length of one side, the perimeter is $4\left(\frac{\mathit{a}}{4}\right)=\mathit{a}$Choice A is incorrect because if the perimeter of one face of the cube is $\frac{\mathit{a}}{4}$,then the total surface area of the cube is $6{\left(\frac{\frac{\mathit{a}}{4}}{4}\right)}^2=6{\left(\frac{\mathit{a}}{16}\right)}^2$,which is not $6{\left(\frac{\mathit{a}}{4}\right)}^2$ . Choice C is incorrect because if the perimeter of one face of the cube is 4a, then the total surface area of the cube is $6{\left(\frac{4\mathit{a}}{4}\right)}^2=6{\mathit{a}}^2$,which is not $6{\left(\frac{\mathit{a}}{4}\right)}^2$ Choice D is incorrect because if the perimeter of one face of the cube is 6a, then the total surface area of the cube is $6{\left(\frac{6\mathit{a}}{4}\right)}^2=6{\left(\frac{3\mathit{a}}{4}\right)}^2$,which is not $6{\left(\frac{\mathit{a}}{4}\right)}^2$