GMAT Quantitative - Cubes

What is the length of the edge of a cube?

Its volume is 1,728 cubic meters.
Its surface area is 864 square meters

EACH statement ALONE is sufficient.

Statement 1 ALONE is sufficient, but Statement 2 alone is not sufficient.

Statement 2 ALONE is sufficient, but Statement 1 alone is not sufficient.

Statements 1 and 2 TOGETHER are not sufficient.

BOTH statements TOGETHER are sufficient, but neither statement ALONE is sufficient.

Explanation

Call the sidelength, surface area, and volume of the cube , , and , respectively.

Then

$V = s^{3}$

or, equivalently,

$s = \sqrt[3]{V}$

So, given statement 1 alone - that is, given only the volume, you can demonstrate the sidelength to be

$s = \sqrt[3]{1,728} = 12$

Also,

$A = 6s^{2}$

or, equivalently,

$s = \sqrt{\frac{A}{6}}$

Given statement 2 alone - that is, given only the surface area, you can demonstrate the sidelength to be

$s = \sqrt{\frac{864}{6}} = \sqrt{144} = 12$

Therefore, the answer is that either statement alone is sufficient.

Ron is making a box in the shape of a cube. He needs to know how much wood he needs. Find the surface area of the box.

I) The diagonal distance across the box will be equivalent to $4\sqrt{3}ft$ .

II) Half the length of one side is .

Either statement is sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Both statements are needed to answer the question.

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.

Explanation

To find the surface area of a cube, we need the length of one side.

Statement I gives the diagonal, we can use this to find the length of one side.

Statement II gives us a clue about the length of one side; we can use that to find the full length of one side.

The following formula gives us the surface area of a cube:

Use Statement I to find the length of the side with the following formula, where is the diagonal and is the side length:

$d=\sqrt{3}a$

$4\sqrt{3}=\sqrt{3}a$

So, using Statement I, we find the surface area to be

$SA=6\cdot 4^2=6\cdot 16=84 ft^2$

Using Statement, we get that the length of one side is two times two:

$2\cdot 2=4ft$

Again, use the surface area formula to get the following:

$SA=6\cdot 4^2=6\cdot 16=84 ft^2$

What is the length of edge of cube ?

(1) $HB=6\sqrt{3}$ .

(2) $\measuredangle {EHB}=45^{\circ}$ .

Statement 1 alone is sufficient.

Statement 2 alone is sufficient.

Both statements together are sufficient.

Each statement alone is sufficient.

Statements 1 and 2 together are not sufficient.

Explanation

In order to find the length of an edge, we would need any information about one of the faces of the cube or about the diagonal of the cube.

Statement 1 gives us the length of the diagonal of the cube, since the formula for the diagonal is $d=s\sqrt{3}$ where is the length of an edge of the cube and is the length of the diagonal we are able to find the length of the edge. Therefore statement 1 alone is sufficient.

Statement 2 alones is insufficient, it gives us something we can already tell knowing that ABCDEFGH is a cube.

Statement 1 alone is sufficient.

is a cube. What is the length of edge ?

(1) The volume of the cube is .

(2) The area of face is .

Each statement alone is sufficient.

Statement 1 alone is sufficient.

Statement 2 alone is sufficient.

Both statements together are sufficient.

Statements 1 and 2 together are not sufficient.

Explanation

Like we have previously seen, to find the length of an edge, we need to have information about the other faces or anything else within the cube.

Statement 1 tells us that the volume of the cube is , from this we can find the length of the side of the cube. Statement 1 alone is sufficient.

Statement 2, tells us that the area of ABCD is , similarily, by taking the square root of this number, we can find the length of the edge of the cube.

Therefore each statement alone is sufficient.

Find the volume of the cube.

1. The cube has a diagonal of 17.32 inches.

2. The cube has a surface area of 600 square inches.

Each statement alone is sufficient.

statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Statements 1 and 2 together are not sufficient.

Explanation

To find the volume of a cube, we only need the length of one side. Using statement 1, we can figure out the length of a side based on the diagonal. We can use the figure below to find the ratio of the diagonal to one side. If the we let the length of one side be x, we can use Pythagorean's theorem to find the length of the diagonal. So, in triangle BCD, we have a right triangle, with two sides of length x. We can set up the equation that the length of BD is $\sqrt{x^2+x^2}= \sqrt{2x^2}=x\sqrt2$ .

Then we can see triangle ADB is also a right triangle. Using Pythagorean's theorem we get the length of AB is $\sqrt{x^2 + (x\sqrt2)^2} = \sqrt{x^2+2x^2}= \sqrt{3x^2} = x\sqrt3$ .

Cube

So, if we divide the number from statement 1 by the square root of 3, we get the length of each side of the cube. Doing this, we get $\frac{17.32}{\sqrt3}= 10$ . Thus, we can solve this problem with just the information from statement 1.

Now, we can also check statement 2. If we know the surface area of the cube, we can use that information to find the length of each side of the cube. We know that the surface area of a cube is the sum of the six faces of the cube, which all have equal area and are all squares. We can divide the total surface area by 6 to find the surface area of each square face. So, 600/6 = 100. We know that the area of a square is just the length of one side squared, so we can take the square root of 100 to find that the length of each side is 10. Thus statement 2 is also sufficient to solve this problem.

Therefore, the answer is that either statement alone is sufficient to answer the question.

A sphere is inscribed inside a cube. What is the volume of the sphere?

Statement 1: The surface area of the cube is 216.

Statement 2: The volume of the cube is 216.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation

The diameter of a sphere inscribed inside a cube is equal to the length of one of the edges of a cube. From either the surface area or the volume of a cube, the appropriate formula can be used to calculate this length. Half this is the radius, from which the formula $V = \frac{4}{3} \pi r^{3}$ can be used to find the volume of the sphere.

Give the length of the diagonal of the cube.

The surface area of the cube is .
The volume of the cube is .

Each statement alone is sufficient to answer the question.

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

Explanation

Statement 1: To find the length of the diagonal of the cube we need to find the length of the cube's edge. We can use the given surface area value to do so:

Now that we know the length of the cube's edge, we can calculate the diagonal:

$d=a\sqrt{3}=7\sqrt{3}\approx 12.12cm$

Statement 2: We're given the volume of the cube which we can also use to solve for the length of the cube's edge.

$a=\sqrt[3]{343}=7$

Knowing the length of the cube's edge allows us to calculate the diagonal:

$d=a\sqrt{3}=7\sqrt{3}\approx 12.12cm$

Each statement alone is sufficient to answer the question.

Calculate the volume of a cube.

The length of the cube's edge is .
The surface area of the cube is .

Each statement alone is sufficient to answer the question.

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

Explanation

Statement 1: We need the length of the edge of the cube to calculate the volume. In this statement we're provided with the length so we just need to plug it into the equation for a cube's volume.

Statement 2: In this case, we need to solve for the length of the cube's edge which we can easily do:

Now that we have our length, we can calculate the volume.

Find the volume of cube W.

I) The base of W has an area of leagues squared.

II) The diagonal of the base of W has a length of $12\sqrt{2}$ leagues.

Each statement alone is enough to solve the question.

Statement 1 is sufficient to solve the question, but statement 2 is not sufficient to solve the question.

Statement 2 is sufficient to solve the question, but statement 1 is not sufficient to solve the question.

Neither statement is sufficient to solve the question. More information is needed.

Both statements taken together are sufficient to solve the question.

Explanation

If we know the base of W, we can find the side length. We then cube the side length to find the volume of a square.

If we know the diagonal of the base of W, we can find the side length. As outlined above, we can then cube the side length to find the volume.

Therefore either statement alone is sufficient to solve the question.

Find the length of an edge of the cube.

The volume of the cube is .
The surface area of the cube is .

Each statement alone is sufficient to answer the question.

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

Explanation

Statement 1: Use the volume formula for a cube to solve for the side length.

where represents the length of the edge

$a=\sqrt[3]{125}=5cm$

Statement 2: Use the surface area formula for a cube to solve for the side length.

Each statement alone is sufficient to answer the question.

Cubes

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