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High School Math : Cubes

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

The side of a cube has a length of $\dpi{100} \small 5 cm$ . What is the total surface area of the cube?

Possible Answers:

$\dpi{100} \small 5cm^{2}$

$\dpi{100} \small 150cm^{2}$

$\dpi{100} \small 50 cm^{2}$

$\dpi{100} \small 25 cm^{2}$

$\dpi{100} \small 125 cm^{2}$

Correct answer:

$\dpi{100} \small 150cm^{2}$

Explanation:

A cube has 6 faces. The area of each face is found by squaring the length of the side.

$\dpi{100} \small 5\times 5 = 25$

Multiply the area of one face by the number of faces to get the total surface area of the cube.

$\dpi{100} \small 25 \times 6=150$

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Example Question #2 : How To Find The Surface Area Of A Cube

What is the surface area of a cube if its height is 3 cm?

Possible Answers:

$45\ cm^2$

$63\ cm^2$

$36\ cm^2$

$54\ cm^2$

$25\ cm^2$

Correct answer:

$54\ cm^2$

Explanation:

The area of one face is given by the length of a side squared.

$A_{face}=(3cm)^2=9\ cm^2$

The area of 6 faces is then given by six times the area of one face: 54 cm².

$A_{total}=6(A_{face})=6(9\ cm^2)=54\ cm^2$

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

A sphere with a volume of $\frac{32}{3}$ $\pi m^{3}$ is inscribed in a cube, as shown in the diagram below.

Act4

What is the surface area of the cube, in $m^{2}$ ?

Possible Answers:

$48\pi \ m^{2}$

$96\ m^{2}$

$24\ m^{2}$

$48\ m^{2}$

$48\pi ^{2}\ m^{2}$

Correct answer:

$96\ m^{2}$

Explanation:

We must first find the radius of the sphere in order to solve this problem. Since we already know the volume, we will use the volume formula to do this.

$V_{sphere}=\frac{4}{3}\pi r^{3}$

$\frac{32}{3}\pi =\frac{4}{3}\pi r^{3}$

$\frac{32}{3}=\frac{4}{3}r^{3}$

$\frac{3}{4}\cdot \frac{32}{3}= r^{3}$

$8=r^{3}$

r=2

With the radius of the sphere in hand, we can now apply it to the cube. The radius of the sphere is half the distance from the top to the bottom of the cube (or half the distance from one side to another). Therefore, the radius represents half of a side length of a square. So in this case

$side=2\cdot 2=4$

The formula for the surface area of a cube is:

$SA_{cube}=6s^{2}$

$6s^{2}=6\cdot (4)^{2}=6\cdot 16=96$

The surface area of the cube is $96\ m^{2}$

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Example Question #11 : How To Find The Surface Area Of A Cube

If a right triangle has a hypotenuse of length 5, and the length of the other sides are and , what would be the surface area of a cube having side length ?

Possible Answers:

$5\sqrt{5}$

None of these answers.

$25\sqrt{5}$

Correct answer:

Explanation:

By the Pythagorean Theorem,

x^2 + (2x)^2 = 5^2

x^2 + 4x^2 = 25

5x^2 = 25

x^2 = 5

$x=\sqrt{5}$

The surface area of a cube having 6 sides, is 6 times the area of one of its sides.

The area of any side of a cube is the square of the side length.

So if the side length is , the area of any side is x^2 , or .

Thus the surface area of the cube is

6*(5)

=30

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

A box measures $11 \ cm \times 10 \ cm \times 8 \ cm$ . How many dice can fit in this box if the dice are cubes with sides of length $2 \ cm$ ?

Possible Answers:

dice

100 dice

110 dice

800 dice

880 dice

Correct answer:

100 dice

Explanation:

Since the dice are $2 \ cm$ on each of their sides, of them would measure $10 \ cm$ in length when standing face-to-face. This means dice will fit along the edge of the box that measures $10 \ cm$ .

dice will also fit on the edge of the box measuring $11 \ cm$ , but will not fit since adding an additional die would bring the length of the dice standing face-to-face up to $12 \ cm$ . 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 $8 \ cm$ .

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 $\times$ dice $\times$ 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:

$5 \times 5 \times 4$ dice = 100 dice.

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Example Question #2 : How To Find The Volume Of A Cube

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

Possible Answers:

343

300

Correct answer:

343

Explanation:

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 $Volume\: of\: Cube=s^{3}$

Using this formula we plug in the side length for to get $V=7^{3}$

Cube the side length to arrive at the answer of $7^{3}=343$

The answer is 343 .

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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:

843.75

Correct answer:

Explanation:

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

We plug the volume into the equation yielding $3375= s^{3}$

Then we take the cube root of both sides giving us $\sqrt[3]{3375}=\sqrt[3]{s^{3}}$

We now know 15=s

The answer is .

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Example Question #21 : Cubes

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

Possible Answers:

None of the answers are correct

Correct answer:

Explanation:

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

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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 l=2.4cm , a width of w=255mm , and a height of h=0.44m ?

Possible Answers:

$2448.01cm^{3}$

$2692.8cm^{3}$

$\dpi{100} 2962.8cm^{3}$

$26928cm^{3}$

$\dpi{100}269.28 cm^{3}$

Correct answer:

$2692.8cm^{3}$

Explanation:

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.

$w=\frac{255mm}{1}*\frac{1cm}{10mm}$

w=25.5cm

Solve for the height.

$h=\frac{0.44m}{1}*\frac{100cm}{1m}$

h=44cm

Now, solve for volume by using the formula V=l*w*h .

V=2.4cm*25.5cm*44cm

$\rightarrow 2692.8cm^{3}$

If you calculated $\dpi{100}269.28 cm^{3}$ , then you did not convert into appropriate units.

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Example Question #5 : How To Find The Volume Of A Cube

Square_with_diagonal What is the volume of a cube with a diagonal one one of its faces of $\dpi{100} d=6.27ft$ ?

Possible Answers:

$246.50ft^{3}$

$87.15ft^{3}$

Not enough information to solve

$264.50ft^{3}$

$78.51ft^{3}$

Correct answer:

$87.15ft^{3}$

Explanation:

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: $1:\sqrt{2}$ .

$\frac{s}{d}=\frac{1}{\sqrt{2}}$

$\dpi{100} \dpi{100} \dpi{100} \frac{s}{6.27ft}=\frac{1}{\sqrt{2}}$

Rearrange an solve for .

$\dpi{100} \dpi{100} s=\frac{6.27ft}{\sqrt{2}}$

Now, solve for the volume.

$V=s^{3}$

$V=(\frac{6.27ft}{\sqrt{2}})*(\frac{6.27ft}{\sqrt{2}})*(\frac{6.27ft}{\sqrt{2}})$

$V=\frac{246.491883ft^{3}}{2\sqrt{2}}$

$\rightarrow87.15ft^{3}$

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High School Math : Cubes

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

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

Example Question #2 : How To Find The Surface Area Of A Cube

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

Example Question #11 : How To Find The Surface Area Of A Cube

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

Example Question #2 : How To Find The Volume Of A Cube

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

Example Question #21 : Cubes

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

Example Question #5 : How To Find The Volume Of A Cube

All High School Math Resources

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