SAT Mathematics Practice Test 5

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