Solid Geometry - GRE Quantitative Reasoning

Card 0 of 20

Question

The surface area of a cube is 486 units. What is the distance of its diagonal (e.g. from its front-left-bottom corner to its rear-right-top corner)?

Answer

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)

Compare your answer with the correct one above

Question

You have a rectangular box with dimensions 6 inches by 6 inches by 8 inches. What is the length of the shortest distance between two non-adjacent corners of the box?

Answer

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 $\dpi{100} \small 6\sqrt{2}$ .

Compare your answer with the correct one above

Question

What is the length of the diagonal of a cube with side lengths of each?

Answer

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:

$\sqrt{s^2+s^2+s^2}$ , or $\sqrt{3s^2}$ , or $s\sqrt{3}$

Now, if the the value of is , we get simply $4\sqrt{3}$

Compare your answer with the correct one above

Question

What is the length of the diagonal of a cube that has a surface area of ?

Answer

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:

$\sqrt{s^2+s^2+s^2}$ , or $\sqrt{3s^2}$ , or $s\sqrt{3}$

Now, if the the value of is , we get simply $11\sqrt{3}$

Compare your answer with the correct one above

Question

A cube with a surface area of 216 square units has a side length that is equal to the diameter of a certain sphere. What is the surface area of the sphere?

Answer

Begin by solving for the length of one side of the cube. Use the formula for surface area to do this:

s= length of one side of the cube

The length of the side of the cube is equal to the diameter of the sphere. Therefore, the radius of the sphere is 3. Now use the formula for the surface area of a sphere:

$Radius=\frac{Diameter}{2}=3$

$SA=4\pi r^2$

$=4\pi(3)^2$

$=4\pi(9)$

$=36\pi$

The surface area of the sphere is $36\pi$ .

Compare your answer with the correct one above

Question

The surface area of a sphere is $36\pi$ . What is its diameter?

Answer

The surface area of a sphere is defined by the equation:

$SA = 4\pi r^2$

For our data, this means:

$36\pi = 4\pi r^2$

Solving for , we get:

or

The diameter of the sphere is .

Compare your answer with the correct one above

Question

The volume of one sphere is $432\pi x^3$ . What is the diameter of a sphere of half that volume?

Answer

Do not assume that the diameter will be half of the diameter of a sphere with volume of $432\pi x^3$ . Instead, begin with the sphere with a volume of $216\pi x^3$ . Such a simple action will prevent a vexing error!

Thus, we know from our equation for the volume of a sphere that:

$216 \pi x^3=\frac{4}{3}\pi r^3$

Solving for , we get:

If you take the cube-root of both sides, you have:

$r = \sqrt[3]{162x^3}$

First, you can factor out an :

$r = x\sqrt[3]{162}$

Next, factor the :

$r = x\sqrt[3]{2*3^4}$

Which simplifies to:

$r = 3x\sqrt[3]{6}$

Thus, the diameter is double that or:

$6x\sqrt[3]{6}$

Compare your answer with the correct one above

Question

Quantity A: The length of a side of a cube with a volume of .

Quantity B: The length of a side of a cube with surface area of .

Which of the following is true?

Answer

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.

Compare your answer with the correct one above

Question

What is the length of an edge of a cube with a surface area of ?

Answer

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

Compare your answer with the correct one above

Question

If a cube has a total surface area of square inches, what is the length of one edge?

Answer

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.

Compare your answer with the correct one above

Question

The surface area of a rectangular prism is , and the lengths of two sides are and . What is the volume of the prism?

Answer

The surface area of a rectangular prism with sides , , and is given as:

.

Two sides are known; it does not matter how they are designated, but for this problem let and , with as the unknown side. This yields equality:

Now that the three dimensions are known, it's possible to calculate the volume:

Compare your answer with the correct one above

Question

What is the radius of a sphere with volume $36\pi$ cubed units?

Answer

The volume of a sphere is represented by the equation $\frac{4}{3}\pi r^3$ . Set this equation equal to the volume given and solve for r:

$36\pi=\frac{4}{3}\pi r^3$

$36\pi\cdot \frac{3}{4}=\pi r^3$

$27\pi=\pi r^3$

Therefore, the radius of the sphere is 3.

Compare your answer with the correct one above

Question

If a sphere has a volume of cubic inches, what is the approximate radius of the sphere?

Answer

The formula for the volume of a sphere is

$v=\frac{4}{3}(\pi r^3)$ where is the radius of the sphere.

Therefore,

$r=\sqrt[3]{\frac{3v}{(4\pi)}}$

$r=\sqrt[3]{\frac{3(268.08)}{(4\pi)}}=\sqrt[3]{63.999}=3.999$ , giving us $r\approx4$ .

Compare your answer with the correct one above

Question

A rectangular prism has the dimensions $9\textup{ x }6\textup{ x }12$ . What is the volume of the largest possible sphere that could fit within this solid?

Answer

For a sphere to fit into the rectangular prism, its dimensions are constrained by the prism's smallest side, which forms its diameter. Therefore, the largest sphere will have a diameter of , and a radius of .

The volume of a sphere is given as:

$\frac{4}{3}\pi r^3$

And thus the volume of the largest possible sphere to fit into this prism is

$\frac{4}{3}\pi (3)^3=36\pi$

Compare your answer with the correct one above

Question

A rectangular box is 6 feet wide, 3 feet long, and 2 feet high. What is the surface area of this box?

Answer

The surface area formula we need to solve this is 2_ab_ + 2_bc_ + 2_ac_. So if we let a = 6, b = 3, and c = 2, then surface area:

= 2(6)(3) + 2(3)(2) +2(6)(2)

= 72 sq ft.

Compare your answer with the correct one above

Question

What is the surface area of a rectangular box that is 3 feet high, 6 feet long, and 4 feet wide?

Answer

Surface area of a rectangular solid

= 2_lw_ + 2_lh_ + 2_wh_

= 2(6)(4) + 2(6)(3) + 2(4)(3)

= 108

Compare your answer with the correct one above

Question

A large cube is made by fitting 8 smaller, identical cubes together. If the volume of each of the smaller cubes is 27, what is the surface area of the large cube?

Answer

Since the volume of the smaller cubes with edges, , is 27, we have:

$V= (width)(length)(height)=e^3=27\Rightarrow e=3$ .

The large cube has edges .

So the surface area of the large cube is:

.

Compare your answer with the correct one above

Question

Quantity A:

The surface area of a cube with a volume of .

Quantity B:

The volume of a cube with a surface area of .

Answer

The relationship can be determined, because it is possible to find the surface area of a cube from the volume and vice versa.

Quantity A:

To find the surface area of the cube, you must find the side length. To find the side length from the volume, you must find the cube root.

Find the cube root of the volume.

$\sqrt[3]{729} = 9$

Insert into surface area equation.

$SA = 6s^{2}$

$SA = 6(9)^{2}$

Quantity B:

To find the volume of a cube, you must find the side length. To find the side length from the surface area, you must divide by 6, then find the square root of the result. Then, cube that result.

$384 = 6s^{2}$

Divide by 6.

$s^{2} = 64$

Square root.

Now, to find the volume.

$V = s^{3}$

$V = 8^{3}$

Quantity B is greater.

Compare your answer with the correct one above

Question

The area of the base of a circular right cylinder is quadrupled. By what percentage is the outer face increased by this change?

Answer

The base of the original cylinder would have been πr2, and the outer face would have been 2πrh, where h is the height of the cylinder.

Let's represent the original area with A, the original radius with r, and the new radius with R: therefore, we know πR2 = 4A, or πR2 = 4πr2. Solving for R, we get R = 2r; therefore, the new outer face of the cylinder will have an area of 2πRh or 2π2rh or 4πrh, which is double the original face area; thus the percentage of increase is 100%. (Don't be tricked into thinking it is 200%. That is not the percentage of increase.)

Compare your answer with the correct one above

Question

A cylinder has a radius of 4 and a height of 8. What is its surface area?

Answer

This problem is simple if we remember the surface area formula!

$SA = 2\pi r^2 +2\pi rh = 2\pi 16+2\pi 4*8 = 32\pi +64\pi = 96\pi$

Compare your answer with the correct one above

Tap the card to reveal the answer