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Intermediate Geometry : How to find the surface area of a cube

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Example Questions

Example Question #22 : Cubes

One side of a cube is $6\:units$ $\displaystyle 6\:units$ long. What is the surface area of the cube?

Possible Answers:

$180\:units^2$ $\displaystyle 180\:units^2$

$216\:units^2$ $\displaystyle 216\:units^2$

$6\:units^2$ $\displaystyle 6\:units^2$

$200\:units^2$ $\displaystyle 200\:units^2$

$36\:units^2$ $\displaystyle 36\:units^2$

Correct answer:

$216\:units^2$ $\displaystyle 216\:units^2$

Explanation:

To find the surface area of a cube, we find the area of a face by multiplying two of its sides together; then, we multiply by $\displaystyle 6$ , since a cube has six faces. So, if $\displaystyle a$ is the length of one side of a cube, then the cube's surface area can be represented as 6a^2 $\displaystyle 6a^2$ .

We know that for this problem, $a=6\:units$ $\displaystyle a=6\:units$ , so we can substitute this value into the equation and solve for the cube's surface area:

$SA=6*a^2=6*6^2=6*36=216\:units^2$ $\displaystyle SA=6*a^2=6*6^2=6*36=216\:units^2$

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Example Question #31 : Cubes

The volume of a cube is $\displaystyle 512 cm^3$ .

What is the surface area of the cube?

Possible Answers:

256 $\displaystyle 256$ cm^2 $\displaystyle cm^2$

512 $\displaystyle 512$ cm^2 $\displaystyle cm^2$

384 $\displaystyle 384$ cm^2 $\displaystyle cm^2$

768 $\displaystyle 768$ cm^2 $\displaystyle cm^2$

171 $\displaystyle 171$ cm^2 $\displaystyle cm^2$

Correct answer:

384 $\displaystyle 384$ cm^2 $\displaystyle cm^2$

Explanation:

Using the volume given, we take it's cube to find the length of the cube: $^3\sqrt{512}=8$ $\displaystyle ^3\sqrt{512}=8$ 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: $8\cdot8\cdot6=384 cm^2$ $\displaystyle 8\cdot8\cdot6=384 cm^2$ .

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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 $1\:cm$ $\displaystyle 1\:cm$ , what is the cube's surface area?

Possible Answers:

$216\:cm^2$ $\displaystyle 216\:cm^2$

$50\:cm^2$ $\displaystyle 50\:cm^2$

$1\:cm^2$ $\displaystyle 1\:cm^2$

$36\:cm^2$ $\displaystyle 36\:cm^2$

$6\:cm^2$ $\displaystyle 6\:cm^2$

Correct answer:

$6\:cm^2$ $\displaystyle 6\:cm^2$

Explanation:

Write the formula to find the surface of a cube, where $\displaystyle x$ is the length.

A=6x^2 $\displaystyle A=6x^2$

Substitute and solve.

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

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Example Question #31 : Cubes

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

Possible Answers:

$3\sqrt2\:in^2$ $\displaystyle 3\sqrt2\:in^2$

$6\:in^2$ $\displaystyle 6\:in^2$

$6\sqrt2\:in^2$ $\displaystyle 6\sqrt2\:in^2$

$24\:in^2$ $\displaystyle 24\:in^2$

$12\:in^2$ $\displaystyle 12\:in^2$

Correct answer:

$12\:in^2$ $\displaystyle 12\:in^2$

Explanation:

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

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

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Example Question #32 : Cubes

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

Possible Answers:

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

$\displaystyle x^3+9x^2+27x+27$

6x+18 $\displaystyle 6x+18$

$\displaystyle 3x^2+18x+27$

$\displaystyle 6x^2+36x+54$

Correct answer:

$\displaystyle 6x^2+36x+54$

Explanation:

The formula for the surface area of a cube is:

SA=6s^2 $\displaystyle SA=6s^2$ , where $\displaystyle s$ 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:

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

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Example Question #32 : Cubes

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

Possible Answers:

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

None of these

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

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

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

Correct answer:

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

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:

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

Total surface area:

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

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Example Question #31 : Solid Geometry

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

Possible Answers:

$49\;cm^2$ $\displaystyle 49\;cm^2$

$45\;cm^2$ $\displaystyle 45\;cm^2$

$54\;cm^2$ $\displaystyle 54\;cm^2$

$9\;cm^2$ $\displaystyle 9\;cm^2$

$27\;cm^2$ $\displaystyle 27\;cm^2$

Correct answer:

$54\;cm^2$ $\displaystyle 54\;cm^2$

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:

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

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

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

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Intermediate Geometry : How to find the surface area of a cube

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #22 : Cubes

Example Question #31 : Cubes

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

Example Question #31 : Cubes

Example Question #32 : Cubes

Example Question #32 : Cubes

Example Question #31 : Solid Geometry

All Intermediate Geometry Resources

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