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High School Math : Solid Geometry

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Example Question #6 : How To Find The Surface Area Of A Cube

Square_with_diagonal What is the surface area of a cube with a diagonal of $\dpi{100} d=4.2cm$ ?

Possible Answers:

$42.22cm^{2}$

$59.22cm^{2}$

$52.92cm^{2}$

$105.84cm^{2}$

$84.05cm^{2}$

Correct answer:

$52.92cm^{2}$

Explanation:

A few facts need to be known to solve this problem. Observe that the diagonal of the square face of the cube cuts it 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: $1:\sqrt{2}$ .

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

$\dpi{100} \dpi{100} \frac{s}{4.2cm}=\frac{1}{\sqrt{2}}$

Rearrange an solve for .

$\dpi{100} s=\frac{4.2cm}{\sqrt{2}}$

Now, solve for the area of the cube using the formula $\dpi{100} \dpi{100} SA=6(s^{2})$ .

$SA=6*(\frac{4.2cm}{\sqrt{2}})^{2}$

$SA=\frac{6}{1}*(\frac{4.2cm}{\sqrt{2}}*\frac{4.2cm}{\sqrt{2}})$

$\dpi{100} SA=\frac{6}{1}*(\frac{17.64cm^{2}}{2})$

$\dpi{100} \dpi{100} SA=3*17.64cm^{2}$

$\rightarrow 52.92cm^{2}$

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

This figure is a cube with one face having an area of 16 in². Cube

What is the surface area of the cube (in²)?

Possible Answers:

256

Correct answer:

Explanation:

The surface area of a cube is the sum of the area of each face. Since there are 6 faces on a cube, the surface area of the entire cube is $6\cdot16$ .

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Example Question #11 : Cubes

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

Possible Answers:

Correct answer:

Explanation:

To find the surface area of a cube, square the length of one edge and multiply the result by six: $6(a^{2})$

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

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Example Question #9 : How To Find The Surface Area Of A Cube

The side length of a particular cube is $\frac{3}{2}$ . What is the surface area of this cube?

Possible Answers:

$\frac{9}{2}$

$\frac{27}{2}$

Correct answer:

$\frac{27}{2}$

Explanation:

To find the surface of a cube, use the standard equation:

SA=6a^2

where denotes the side length.

Plug in the given value for to find the answer:

$SA=6\cdot \left(\frac{3}{2}\right)^2=6\cdot \left(\frac{9}{4}\right)=\frac{27}{2}$

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Example Question #10 : How To Find The Surface Area Of A Cube

Sarah is wrapping a birthday present. The box is a cube with sides of $6\; in$ . At a minimum, how many square feet of wrapping paper will she need?

Possible Answers:

$1.00\; ft^{2}$

$0.50\; ft^{2}$

$1.50\; ft^{2}$

$0.75\; ft^{2}$

$1.25\; ft^{2}$

Correct answer:

$1.50\; ft^{2}$

Explanation:

Remember, $6\; in=0.50\; ft$ .

For a cube:

$SA = 6s^{2}$

Thus $6(0.50)^{2}=6(0.25)=1.50\; ft^{2}$ .

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Example Question #11 : Cubes

$\textup{What is the surface area, in terms of }p,\textup{of a cube with sides of length }p\textup{ ?}$

Possible Answers:

12p

$p^{2}$

$4p^{2}$

$p^{3}$

$6p^{2}$

Correct answer:

$6p^{2}$

Explanation:

$\textup{Each of the six identical faces of the cube is a square with sides }p.$

$\textup{Each face: }A=p^{2}\: \: \: \: \: \textup{Surface area}=6p^{2}$

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Example Question #581 : Geometry

Find the surface area of the following cube:

Length_of_diagonal

Possible Answers:

$80\ m^2$

$96\ m^2$

$90\ m^2$

$82\ m^2$

$88\ m^2$

Correct answer:

$96\ m^2$

Explanation:

The formula for the surface area of a cube is

SA=6(s)^2 ,

where is the length of the side.

Plugging in our values, we get:

$SA=6(4\ m)^2$

$SA=96\ m^2$

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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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High School Math : Solid Geometry

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #6 : How To Find The Surface Area Of A Cube

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

Example Question #11 : Cubes

Example Question #9 : How To Find The Surface Area Of A Cube

Example Question #10 : How To Find The Surface Area Of A Cube

Example Question #11 : Cubes

Example Question #581 : Geometry

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

All High School Math Resources

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