Phil is making a number cube out of cardboard . He traces one face of the number cube onto a coordinate grid . The coordinates of two opposite vertices of the face are (4, 8) and (12, 16) . What is the surface area, in square units, of Phil’s entire number cube?

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384 To find the surface area of the entire number cube, we need to find the area of each face and sum them all together. Since opposite faces of a cube have the same area, we only need to find the area of one face and then double it.

To find the area of a face, we need to find the length of each side. In this case, we have two opposite vertices of the face, which allows us to find the length of each side.

Let's start by finding the length of one side:

The x-coordinate of the first vertex is 4, and the x-coordinate of the second vertex is 12. So the length of one side in the x-direction is 12 - 4 = 8 units.

Similarly, the y-coordinate of the first vertex is 8, and the y-coordinate of the second vertex is 16. So the length of one side in the y-direction is 16 - 8 = 8 units.

Since the square has equal sides, the length of one side is 8 units.

Now that we know the length of one side, we can find the area of one face by squaring this length:

Area of one face = (side length)^2 = 8^2 = 64 square units.

Since the cube has 6 faces of equal area, the surface area of the entire cube is 6 times the area of one face:

Surface area of the entire cube = 6 * 64 = 384 square units.

Therefore, the surface area of Phil's entire number cube is 384 square units.

the diagonal has length

√(8^2+8^2) = 8√2
So, the sides have length 8.

The total area is thus 6*8^2