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

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About mvtrinh

Retired high school math teacher.
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