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High School Math : Geometry

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8 Diagnostic Tests 613 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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

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

Possible Answers:

$\displaystyle 80$

$\displaystyle 64$

$\displaystyle 96$

$\displaystyle 16$

$\displaystyle 8$

Correct answer:

$\displaystyle 96$

Explanation:

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

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

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Example Question #1891 : High School Math

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

Possible Answers:

$\displaystyle 9$

$\frac{27}{2}$ $\displaystyle \frac{27}{2}$

$\displaystyle 27$

$\frac{9}{2}$ $\displaystyle \frac{9}{2}$

$\displaystyle 18$

Correct answer:

$\frac{27}{2}$ $\displaystyle \frac{27}{2}$

Explanation:

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

SA=6a^2 $\displaystyle SA=6a^2$

where $\displaystyle a$ denotes the side length.

Plug in the given value for $\displaystyle a$ to find the answer:

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

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Example Question #17 : Cubes

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

Possible Answers:

$1.00\; ft^{2}$ $\displaystyle 1.00\; ft^{2}$

$1.50\; ft^{2}$ $\displaystyle 1.50\; ft^{2}$

$0.75\; ft^{2}$ $\displaystyle 0.75\; ft^{2}$

$1.25\; ft^{2}$ $\displaystyle 1.25\; ft^{2}$

$0.50\; ft^{2}$ $\displaystyle 0.50\; ft^{2}$

Correct answer:

$1.50\; ft^{2}$ $\displaystyle 1.50\; ft^{2}$

Explanation:

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

For a cube:

$SA = 6s^{2}$ $\displaystyle SA = 6s^{2}$

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

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Example Question #581 : Geometry

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

Possible Answers:

$6p^{2}$ $\displaystyle 6p^{2}$

$p^{3}$ $\displaystyle p^{3}$

12p $\displaystyle 12p$

$p^{2}$ $\displaystyle p^{2}$

$4p^{2}$ $\displaystyle 4p^{2}$

Correct answer:

$6p^{2}$ $\displaystyle 6p^{2}$

Explanation:

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

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

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Example Question #582 : Geometry

Find the surface area of the following cube:

Length_of_diagonal

Possible Answers:

$82\ m^2$ $\displaystyle 82\ m^2$

$80\ m^2$ $\displaystyle 80\ m^2$

$88\ m^2$ $\displaystyle 88\ m^2$

$90\ m^2$ $\displaystyle 90\ m^2$

$96\ m^2$ $\displaystyle 96\ m^2$

Correct answer:

$96\ m^2$ $\displaystyle 96\ m^2$

Explanation:

The formula for the surface area of a cube is

$\displaystyle SA=6(s)^2$,

where $\displaystyle s$ is the length of the side.

Plugging in our values, we get:

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

$SA=96\ m^2$ $\displaystyle SA=96\ m^2$

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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 $\dpi{100} \small 5 cm$ $\displaystyle \dpi{100} \small 5 cm$. What is the total surface area of the cube?

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

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

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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:

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

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

$25\ cm^2$ $\displaystyle 25\ cm^2$

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

$63\ cm^2$ $\displaystyle 63\ cm^2$

Correct answer:

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

Explanation:

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

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

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

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

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

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

Act4

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

Possible Answers:

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

$96\ m^{2}$ $\displaystyle 96\ m^{2}$

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

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

$24\ m^{2}$ $\displaystyle 24\ m^{2}$

Correct answer:

$96\ m^{2}$ $\displaystyle 96\ m^{2}$

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.

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

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

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

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

$8=r^{3}$ $\displaystyle 8=r^{3}$

r=2 $\displaystyle r=2$

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

$side=2\cdot 2=4$ $\displaystyle side=2\cdot 2=4$

The formula for the surface area of a cube is:

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

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

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

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Example Question #21 : Cubes

If a right triangle has a hypotenuse of length 5, and the length of the other sides are $\displaystyle x$ and $\displaystyle 2x$, what would be the surface area of a cube having side length $\displaystyle x$?

Possible Answers:

$\displaystyle 30$

None of these answers.

$\displaystyle 25$

$5\sqrt{5}$ $\displaystyle 5\sqrt{5}$

$25\sqrt{5}$ $\displaystyle 25\sqrt{5}$

Correct answer:

$\displaystyle 30$

Explanation:

By the Pythagorean Theorem,

$\displaystyle x^2 + (2x)^2 = 5^2$

$\displaystyle x^2 + 4x^2 = 25$

$\displaystyle 5x^2 = 25$

x^2 = 5 $\displaystyle x^2 = 5$

$x=\sqrt{5}$ $\displaystyle x=\sqrt{5}$

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 $\displaystyle x$, the area of any side is x^2 $\displaystyle x^2$, or $\displaystyle 5$.

Thus the surface area of the cube is

6*(5) $\displaystyle 6*(5)$

=30 $\displaystyle =30$

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

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

Possible Answers:

$\displaystyle 88$ dice

800 $\displaystyle 800$ dice

100 $\displaystyle 100$ dice

110 $\displaystyle 110$ dice

880 $\displaystyle 880$ dice

Correct answer:

100 $\displaystyle 100$ dice

Explanation:

Since the dice are $2 \ cm$ $\displaystyle 2 \ cm$ on each of their sides, $\displaystyle 5$ of them would measure $10 \ cm$ $\displaystyle 10 \ cm$ in length when standing face-to-face. This means $\displaystyle 5$ dice will fit along the edge of the box that measures $10 \ cm$ $\displaystyle 10 \ cm$.

$\displaystyle 5$ dice will also fit on the edge of the box measuring $11 \ cm$ $\displaystyle 11 \ cm$, but $\displaystyle 6$ will not fit since adding an additional die would bring the length of the dice standing face-to-face up to $12 \ cm$ $\displaystyle 12 \ cm$. There are no half-dice to fill the gap, so there is a small empty space on this side.

$\displaystyle 4$ dice will fit along the side that measures $8 \ cm$ $\displaystyle 8 \ cm$.

Treating "dice" as the unit of measurement instead of $\displaystyle cm$ and considering only the volume where the dice will fit, one ends up with a rectangular shape measuring $\displaystyle 5$ dice $\times$ $\displaystyle \times$ $\displaystyle 5$ dice $\times$ $\displaystyle \times$ $\displaystyle 4$ 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:

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

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High School Math : Geometry

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

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

Example Question #1891 : High School Math

Example Question #17 : Cubes

Example Question #581 : Geometry

Example Question #582 : 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

Example Question #21 : Cubes

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

All High School Math Resources

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