To find the surface area of a cube means to find the area around the entire object. In the case of a cube, we will need to find that area of all the sides and the top and bottom. Since a cube has equal side lengths, the area of each side and the area of the top and bottom will all be the same.

Recall that the area for a side of a cube is:

From here there are two approaches one can take.

Approach one:

Add all the areas together.

Approach two:

Use the formula for the surface area of a cube,

In this particular case we are given the side length is 3.

Thus we can find the surface area to be,

by approach one,

and by appraoch two,

.

To solve, simply use the formula for the surface area of a cube.

If you do not remember the formula for the test, it is important to draw a picture or to visually conceptualize it. Remember, when finding area of a square or rectangle, you simply multiply the two side lengths. So for a cube in 3 dimensions, you simply have to multiply those three together.

However, a cube is a special case where all three lengths are the same. Thus,

First, we need the side length of the cube. Given that , where is a side length, we can solve for  and set  simply by taking the cube root of both sides.

Then we need to remember or realize that the surface area of a cube is because there will be six identical square faces. Plugging in , we get as our answer.

To solve, simply use the formula for the surface area of a cube.

If this is not a formula you have committed to memory, remember that a cube has 6 faces with equal area. So, start by calculating the surface area of one side (64) and add it 6 times. Thus,

You own a Rubik's cube with a volume of . What is the surface area of the cube?

To solve for edge length, think of the volume of a cube formula:

Now, we have the volume, so just rearrange it to solve for side length:

Next, recall the surface area of a cube formula:

Plug in and simplify to get:

If broken down into parts, this is an easy problem. It is first necessary to isolate the dimensions of the walls. If the room is 10ft high, we know 23 x 17 designates the area of the floor and ceiling. Based on this, we know that the room has the following dimensions for the walls: 23 x 10 and 17 x 10. Since there are two of each, we can calculate the total area of walls - ignoring doors and windows - by doubling the sum of these two areas:

2 * (23 * 10 + 17 * 10) = 2 * (230 + 170) = 2 * 400 = 800 ft²

Now, we merely need to calculate the area "taken out" of the walls:

For the door: 2.5 * 8 = 20 ft² 

For the windows: 3 * 6 = 18 ft²

The total wall space is therefore: 800 – 20 – 18 = 762 ft²

Now, if one can of paint covers 57 ft², we calculate the number of cans necessary by dividing the total exposed area by 57: 762/57 = (approx.) 13.37.

Since we cannot buy partial cans, we must purchase 14 cans.

The volume of a cubic box is given by (side length)³. Thus, the volume of the larger box is (4x)³ = 64x³, while the volume of the smaller box is x³. Divide the volume of the larger box by that of the smaller box, (64x³)/(x³) = 64.

Determine the volume of both cubes and then subtract the smaller from the larger.  The large cube volume is 9” * 9” * 9” = 729 in³ and the small cube is 3” * 3” * 3” = 27 in³.  The difference is 702 in³.

A cube that has three times as long sides is 3x3x3=27 times bigger than the original. Therefore, the answer is 5x27= 135.

Using S as the side length in the original cube, the original is s*s*s. Doubling one side and tripling the other gives 2s*2s*2s for a new volume formula for 8s*s*s, showing that the new volume is 8x greater than the original.