## Surface Area of Cube

Three of the vertices of a cube are $P=\left (7,12,10\right ),Q=\left (8,8,1\right )$ and $R=\left (11,3,9\right )$. What is the surface area of the cube?
Source: NCTM Mathematics Teacher

SOLUTION

Distance formula in 3-dimension of a segment $\overline{AB}$ defined by two end points $A=\left (x_1,y_1,z_1\right )$ and $B=\left (x_2,y_2,z_2\right )$
$AB=\sqrt{\left (x_2-x_1\right )^2+\left (y_2-y_1\right )^2+\left (z_2-z_1\right )^2}$
Apply the formula to segment $\overline{PR}$
$PR=\sqrt{\left (11-7\right )^2+\left (3-12\right )^2+\left (9-10\right )^2}$
$=\sqrt{16+81+1}$
$=\sqrt{98}$
$=7\sqrt{2}$
$\overline{PR}$ is the hypotenuse of an isosceles right triangle of side length $a$; its length equals $a\sqrt{2}$
$a\sqrt{2}=7\sqrt{2}$
$a=7$
The surface area of the cube is made up of the area of the six square faces
$6\left (7^2\right )=294$ square units

Answer: $294$ square units