Cubes - GRE Quantitative Reasoning

The surface area of a cube is merely the sum of the surface areas of the squares that make up its faces. Therefore, the surface area equation understandably is:

, where is the side length of any one side of the cube. For our values, we know:

Solving for , we get:

or

Now, the volume of a cube is defined by the simple equation:

For , this is:

Recall that the volume of a cube is defined by the equation:

, where is the side length of the cube.

Therefore, if we know that , we can solve:

This means that .

Now, if we triple to , the new volume of our cube will be:

Recall that the equation for the surface area of a cube is merely derived from the fact that the cube's faces are made up of squares. It is therefore:

For our values, this is:

Solving for , we get:

, so

Now, the volume of a cube is merely:

Therefore, for , this value is:

To find the volume of a cube, you multiple your side length 3 times (s*s*s).

To find the side length from the volume, you find the cube root which gives you 4

.

Doubling the side gives you 8

.

The volume of the new cube would then be 512

.

The box is 2 times as long as it is high, so H = L/2. It is also 4 times as wide as it is long, so W = 4_L_. Now we need volume = L * W * H = L * 4_L_ * L/2 = 2_L_3.

The surface area of a cube is merely the sum of the surface areas of the squares that make up its faces. Therefore, the surface area equation understandably is:

, where is the side length of any one side of the cube. For our values, we know:

Solving for , we get:

or

Now, the volume of a cube is defined by the simple equation:

For , this is:

Recall that the volume of a cube is defined by the equation:

, where is the side length of the cube.

Therefore, if we know that , we can solve:

This means that .

Now, if we triple to , the new volume of our cube will be:

Recall that the equation for the surface area of a cube is merely derived from the fact that the cube's faces are made up of squares. It is therefore:

For our values, this is:

Solving for , we get:

, so

Now, the volume of a cube is merely:

Therefore, for , this value is:

To find the volume of a cube, you multiple your side length 3 times (s*s*s).

To find the side length from the volume, you find the cube root which gives you 4

.

Doubling the side gives you 8

.

The volume of the new cube would then be 512

.

First, we must ascertain the length of each side. Based on our initial data, we know that the 6 faces of the cube will have a surface area of 6x2. This yields the equation:

6x2 = 486, which simplifies to: x2 = 81; x = 9.

Therefore, each side has a length of 9. Imagine the cube is centered on the origin. This means its "front-left-bottom corner" will be at (–4.5, –4.5, 4.5) and its "rear-right-top corner" will be at (4.5, 4.5, –4.5). To find the distance between these, we use the three-dimensional distance formula:

d = √((x1 – x2)2 + (y1 – y2)2 + (z1 – z2)2)

For our data, this will be:

√( (–4.5 – 4.5)2 + (–4.5 – 4.5)2 + (4.5 + 4.5)2) =

√( (–9)2 + (–9)2 + (9)2) = √(81 + 81 + 81) = √(243) =

√(3 * 81) = √(3) * √(81) = 9√(3)

The shortest length between any two non-adjacent corners will be the diagonal of the smallest face of the rectangular box. The smallest face of the rectangular box is a six-inch by six-inch square. The diagonal of a six-inch square is 6$\sqrt{2}$.

The diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:

, or , or

Now, if the the value of is , we get simply

To begin, the best thing to do is to find the length of a side of the cube. This is done using the formula for the surface area of a cube. Recall that a cube is made up of squares. Therefore, its surface area is:

, where is the length of a side.

Therefore, for our data, we have:

Solving for , we get:

This means that

Now, the diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:

, or , or

Now, if the the value of is , we get simply

Recall that the equation for the volume of a cube is:

Since the sides of a cube are merely squares, the surface area equation is just times the area of one of those squares:

So, for our two quantities:

Quantity A

Use your calculator to estimate this value (since you will not have a square root key). This is .

Quantity B

First divide by :

Therefore,

Therefore, the two quantities are equal.

The surface area of a cube is made up of squares. Therefore, the equation is merely times the area of one of those squares. Since the sides of a square are equal, this is:

, where is the length of one side of the square.

For our data, we know:

This means that:

Now, while you will not have a calculator with a square root key, you do know that . (You can always use your calculator to test values like this.) Therefore, we know that . This is the length of one side

There are 6 sides to a cube. If the total surface area is 54 square inches, then each face must have an area of 9 square inches.

Every face of a cube is a square, so if the area is 9 square inches, each edge must be 3 inches.

Recall that the equation for the volume of a cube is:

Since the sides of a cube are merely squares, the surface area equation is just times the area of one of those squares:

So, for our two quantities:

Quantity A

Use your calculator to estimate this value (since you will not have a square root key). This is .

Quantity B

First divide by :

Therefore,

Therefore, the two quantities are equal.

The surface area of a cube is made up of squares. Therefore, the equation is merely times the area of one of those squares. Since the sides of a square are equal, this is:

, where is the length of one side of the square.

For our data, we know:

This means that:

Now, while you will not have a calculator with a square root key, you do know that . (You can always use your calculator to test values like this.) Therefore, we know that . This is the length of one side

There are 6 sides to a cube. If the total surface area is 54 square inches, then each face must have an area of 9 square inches.

Every face of a cube is a square, so if the area is 9 square inches, each edge must be 3 inches.