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Intermediate Geometry : Solid Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #31 : Cubes

The volume of a cube is 512 cm^3 .

What is the surface area of the cube?

Possible Answers:

512 cm^2

384 cm^2

256 cm^2

768 cm^2

171 cm^2

Correct answer:

384 cm^2

Explanation:

Using the volume given, we take it's cube to find the length of the cube: $^3\sqrt{512}=8$ cm.

Therefore, the length of the cube is 8 cm.

Knowing the properties of a cube, this implies that the width and height of the cube is also 8 cm.

Since all sides are identical, the formula for the surface area is length times width times the number of sides: $8\cdot8\cdot6=384 cm^2$ .

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

If a side of a cube has a length of $1\:cm$ , what is the cube's surface area?

Possible Answers:

$1\:cm^2$

$36\:cm^2$

$6\:cm^2$

$50\:cm^2$

$216\:cm^2$

Correct answer:

$6\:cm^2$

Explanation:

Write the formula to find the surface of a cube, where is the length.

A=6x^2

Substitute and solve.

$A=6(1\:cm)^2=6\:cm^2$

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

Find the surface area of a cube with a side length of $\sqrt2\:in$ .

Possible Answers:

$6\sqrt2\:in^2$

$6\:in^2$

$12\:in^2$

$3\sqrt2\:in^2$

$24\:in^2$

Correct answer:

$12\:in^2$

Explanation:

Write the formula for the surface area of a cube, substitute the length provided in the question, and simplify.

$A=6x^2 = 6(\sqrt2\:in)^2 = 6\cdot2\:in^2 =12\:in^2$

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

if the side length of a cube is (x+3) , what is the cube's surface area?

Possible Answers:

6x^2+36x+54

3x^2+18x+27

$6\sqrt2 (x+3)^2$

x^3+9x^2+27x+27

6x+18

Correct answer:

6x^2+36x+54

Explanation:

The formula for the surface area of a cube is:

SA=6s^2 , where is the length of one side of the cube.

We are given the length of one side of the cube in question, so we can substitute that value into the surface area equation and solve:

SA=6s^2 = 6(x+3)^2 = 6(x^2+6x+9) =6x^2+36x+54

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Example Question #3 : How To Find The Surface Area Of A Cube

A cube has a sphere inscribed inside it with a diameter of 4 meters. What is the surface area of the cube?

Possible Answers:

$164\hspace{1mm}m^2$

$128\hspace{1mm}m^2$

$64\hspace{1mm}m^2$

$96\hspace{1mm}m^2$

None of these

Correct answer:

$96\hspace{1mm}m^2$

Explanation:

Since the sphere is inscribed within the cube, its diameter is the same length as an edge of the cube. Since cubes have identical side lengths we find the area of one side and then multiply by the number of sides to find the total surface area.

Area of one side:

$A=s*s=4*4=16\hspace{1mm}m^2$

Total surface area:

$SA=16\hspace{1mm}m^2*6=\mathbf{96\hspace{1mmm^2}}$

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Example Question #4 : How To Find The Surface Area Of A Cube

A geometric cube has a volume of $27\;cm^3$ . Find the surface area of the cube.

Possible Answers:

$49\;cm^2$

$45\;cm^2$

$9\;cm^2$

$54\;cm^2$

$27\;cm^2$

Correct answer:

$54\;cm^2$

Explanation:

We first need to know the edge length before we can solve for surface area. Since we are provided the volume and all edges are of equal length, we can use the formula for volume to get the length of sides:

$\\volume=length\cdot width\cdot height\\ \\27\;cm^3=a^3\\ \\a=\sqrt[3]{27\;cm^3}\\ \\a=3\;cm$

Now that we know the length of sides, we can plug this value into our surface area formula:

$\\S.A.=6a^2\\S.A.=6(3\;cm)^2\\S.A.=6\cdot9\;cm^2=54\;cm^2$

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

A cube has edges that are three times as long as those of a smaller cube. The volume of the bigger cube is how many times larger than that of the smaller cube?

Possible Answers:

Correct answer:

Explanation:

If we let represent the length of an edge on the smaller cube, its volume is $x^{3}$ .

The larger cube has edges three times as long, so the length can be represented as . The volume is $(3x)^{3}$ , which is $27x^{3}$ .

The large cube's volume of $27x^{3}$ is 27 times as large as the small cube's volume of $x^{3}$ .

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

The side length of a cube is inches.

What is the volume?

Possible Answers:

$343\,in^3$

$294\, in^2$

$21 \, in^3$

$7\, in^3$

Correct answer:

$343\,in^3$

Explanation:

The volume of a cube is

V = s^3

So with a side length of 7 inches, the volume is

$V = 7^{3}\, in^3 = 343\, in^3$

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

The side length of a cube is ft.

What is the volume?

Possible Answers:

$24\, ft^2$

$4\, ft^3$

$6\, ft^3$

$8\,ft^3$

Correct answer:

$8\,ft^3$

Explanation:

The volume of a cube is

V = s^3

So with a side length of 2 ft, the volume is

$V = 2^{3}\, ft^3 = 8\, ft^3$

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Example Question #3 : How To Find The Volume Of A Cube

Find the volume of a cube with an edge length of $7\:in$ .

Possible Answers:

$308\:in^3$

$49\:in^3$

$294\:in^3$

$252\:in^3$

$343\:in^3$

Correct answer:

$343\:in^3$

Explanation:

The volume of a cube can be determined through the equation V = s^3 , where stands for the length of one side of the cube. The equation is $s\cdot s\cdot s$ because all edges in a cube are the same length. The value for the given edge just needs to be substituted into the equation for in order to solve for the cube's volume.

V = s^3

V = 7^3

$V = 7 \cdot 7\cdot 7$

$V = 343\:in^3$

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Intermediate Geometry : Solid Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #31 : Cubes

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

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 #3 : How To Find The Surface Area Of A Cube

Example Question #4 : How To Find The Surface Area Of A Cube

Example Question #1 : How To Find The Volume 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 #3 : How To Find The Volume Of A Cube

All Intermediate Geometry Resources

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