### All High School Math Resources

## Example Questions

### Example Question #51 : Solid Geometry

A box measures . How many dice can fit in this box if the dice are cubes with sides of length ?

**Possible Answers:**

dice

dice

dice

dice

dice

**Correct answer:**

dice

Since the dice are on each of their sides, of them would measure in length when standing face-to-face. This means dice will fit along the edge of the box that measures .

dice will also fit on the edge of the box measuring , but will not fit since adding an additional die would bring the length of the dice standing face-to-face up to . There are no half-dice to fill the gap, so there is a small empty space on this side.

dice will fit along the side that measures .

Treating "dice" as the unit of measurement instead of and considering only the volume where the dice will fit, one ends up with a rectangular shape measuring dice dice dice. The volume of this shape is the number of dice that will fit in this area (and thus the box). Find the volume by multiplying all lengths of the shape's sides together:

dice dice.

### Example Question #51 : Solid Geometry

What is the volume of a cube with a side length of 7?

**Possible Answers:**

**Correct answer:**

When searching for the volume of a cube we are looking for the amount of the space enclosed by the cube.

To find this we must know the formula for the volume of a cube which is

Using this formula we plug in the side length for to get

Cube the side length to arrive at the answer of

The answer is .

### Example Question #1 : How To Find The Volume Of A Cube

If the volume of a cube is 3375, what is the length of each side?

**Possible Answers:**

**Correct answer:**

To find this we must know the formula for the volume of a cube which is

We plug the volume into the equation yielding

Then we take the cube root of both sides giving us

We now know

The answer is .

### Example Question #31 : Solid Geometry

What is the sum of the number of vertices, edges, and faces of a cube?

**Possible Answers:**

24

20

18

None of the answers are correct

26

**Correct answer:**

26

Vertices = three planes coming together at a point = 8

Edges = two planes coming together to form a line = 12

Faces = one plane as the surface of the solid = 6

Vertices + Edges + Faces = 8 + 12 + 6 = 26

### Example Question #1 : How To Find The Volume Of A Cube

What is the volume, in centimeters, of a rectangular prism with a length of , a width of , and a height of ?

**Possible Answers:**

**Correct answer:**

In order to solve this problem, we need to make sure that the measurements given are in uniform units. In this case, we will use centimeters; therefore, we need to convert the other units of meters and millimeters into centimeters via dimensional analysis.

Solve for the width.

Solve for the height.

Now, solve for volume by using the formula .

If you calculated , then you did not convert into appropriate units.

### Example Question #56 : Solid Geometry

What is the volume of a cube with a diagonal one one of its faces of ?

**Possible Answers:**

Not enough information to solve

**Correct answer:**

A few facts need to be known to solve this problem. Observe that the diagonal of the square face of the cube cuts the qradilateral into two right isosceles triangles; therefore, the length of a side of the square to its diagonal is the same as an isosceles right triangle's leg to its hypotenuse: .

Rearrange an solve for .

Now, solve for the volume.

### Example Question #1 : How To Find The Volume Of A Cube

A building is in the shape of a perfect cube, with a side length of . What is the building's volume, in cubic meters?

**Possible Answers:**

Not enough information to solve

**Correct answer:**

In order to solve this question, we need to convert miles to meters via dimensional analysis and then solve for the volume of the building.

Now, solve for volume.

### Example Question #1 : How To Find The Volume Of A Cube

This figure is a cube with one face having an area of 16 in^{2}.

What is the volume of the cube (in^{3})?

**Possible Answers:**

**Correct answer:**

The volume of a cube is one side cubed. Because we know that one face has an area of 16 in^{2}, then we know that one side must be the square root of 16 or 4. Thus the volume is .

### Example Question #1 : How To Find The Volume Of A Cube

A cube has a height of 4 feet. What is the volume of the cube in feet?

**Possible Answers:**

**Correct answer:**

We only have the length of one edge, but that is sufficient for finding the volume. The volume of a cube equals one edge multiplied by itself three times:

### Example Question #1 : How To Find The Volume Of A Cube

A cube has a surface area of units squared. What is its volume?

**Possible Answers:**

units cubed

units cubed

units cubed

units cubed

**Correct answer:**

units cubed

Since a cube has square faces and the surface area of each face is given by multiplying the length of one side of the square face by itself, the equation for the surface area of a cube is , where is the length of one side of one face (i.e., one edge of the cube). Find the length of one side/edge of the given cube by setting the given surface area equal to :

units

The volume of a cube is , where is the length of one of the cube's edges. Substituing the solution to the previous equation for in the volume equation gives the volume of the cube:

units cubed

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