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Intermediate Geometry : Plane Geometry

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Example Questions

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

Find the area of the regular pentagon.

Possible Answers:

1180

1300

1260

1220

Correct answer:

1260

Explanation:

Recall that the area of regular polygons can be found using the following formula:

$\text{Area of Regular Polygon}=\frac{1}{2}(\text{apothem})(\text{perimeter})$

First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:

$\text{Perimeter}=5(\text{side length})$

Substitute in the length of the given side of the pentagon in order to find the perimeter.

$\text{Perimeter}=5(24)$

$\text{Perimeter}=120$

Next, substitute in the given and calculated information to find the area of the pentagon.

$\text{Area of Pentagon}=\frac{1}{2}(21)(120)$

$\text{Area of Pentagon}=1260$

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

Find the area of the regular pentagon.

Possible Answers:

1600

1900

1300

1000

Correct answer:

1600

Explanation:

Recall that the area of regular polygons can be found using the following formula:

$\text{Area of Regular Polygon}=\frac{1}{2}(\text{apothem})(\text{perimeter})$

First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:

$\text{Perimeter}=5(\text{side length})$

Substitute in the length of the given side of the pentagon in order to find the perimeter.

$\text{Perimeter}=5(20)$

$\text{Perimeter}=100$

Next, substitute in the given and calculated information to find the area of the pentagon.

$\text{Area of Pentagon}=\frac{1}{2}(16)(100)$

$\text{Area of Pentagon}=1600$

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Example Question #21 : How To Find The Area Of A Pentagon

Find the area of the regular pentagon.

Possible Answers:

988.5

721.5

1321.5

1092.5

Correct answer:

1092.5

Explanation:

Recall that the area of regular polygons can be found using the following formula:

$\text{Area of Regular Polygon}=\frac{1}{2}(\text{apothem})(\text{perimeter})$

First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:

$\text{Perimeter}=5(\text{side length})$

Substitute in the length of the given side of the pentagon in order to find the perimeter.

$\text{Perimeter}=5(23)$

$\text{Perimeter}=115$

Next, substitute in the given and calculated information to find the area of the pentagon.

$\text{Area of Pentagon}=\frac{1}{2}(19)(115)$

$\text{Area of Pentagon}=1092.5$

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Example Question #21 : How To Find The Area Of A Pentagon

Find the area of the regular pentagon.

Possible Answers:

6630

6790

6120

6530

Correct answer:

6630

Explanation:

Recall that the area of regular polygons can be found using the following formula:

$\text{Area of Regular Polygon}=\frac{1}{2}(\text{apothem})(\text{perimeter})$

First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:

$\text{Perimeter}=5(\text{side length})$

Substitute in the length of the given side of the pentagon in order to find the perimeter.

$\text{Perimeter}=5(39)$

$\text{Perimeter}=195$

Next, substitute in the given and calculated information to find the area of the pentagon.

$\text{Area of Pentagon}=\frac{1}{2}(34)(195)$

$\text{Area of Pentagon}=6630$

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Example Question #801 : Plane Geometry

Find the area of the regular pentagon.

Possible Answers:

Correct answer:

Explanation:

Recall that the area of regular polygons can be found using the following formula:

$\text{Area of Regular Polygon}=\frac{1}{2}(\text{apothem})(\text{perimeter})$

First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:

$\text{Perimeter}=5(\text{side length})$

Substitute in the length of the given side of the pentagon in order to find the perimeter.

$\text{Perimeter}=5(4)$

$\text{Perimeter}=20$

Next, substitute in the given and calculated information to find the area of the pentagon.

$\text{Area of Pentagon}=\frac{1}{2}(2)(20)$

$\text{Area of Pentagon}=20$

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Example Question #22 : How To Find The Area Of A Pentagon

Find the area of the regular pentagon.

Possible Answers:

Correct answer:

Explanation:

Recall that the area of regular polygons can be found using the following formula:

$\text{Area of Regular Polygon}=\frac{1}{2}(\text{apothem})(\text{perimeter})$

First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:

$\text{Perimeter}=5(\text{side length})$

Substitute in the length of the given side of the pentagon in order to find the perimeter.

$\text{Perimeter}=5(6)$

$\text{Perimeter}=30$

Next, substitute in the given and calculated information to find the area of the pentagon.

$\text{Area of Pentagon}=\frac{1}{2}(3)(30)$

$\text{Area of Pentagon}=45$

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Example Question #25 : How To Find The Area Of A Pentagon

Each side of this pentagon has a length of $5\;cm$ .

Solve for the area of the pentagon.

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Possible Answers:

$35.5\;cm^2$

$25\;cm^2$

$43\;cm^2$

$21.5\;cm^2$

$32.7\;cm^2$

Correct answer:

$43\;cm^2$

Explanation:

The formula for area of a pentagon is $A=5\cdot(\frac{1}{2}\cdot s\cdot h)$ , with representing the length of one side and representing the apothem.

To find the apothem, we can convert our one pentagon into five triangles and solve for the height of the triangle:

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Each of these triangles have angle measures of $72$^{\circ}$\; 54$^{\circ}$\;54$^{\circ}$$ , with $72$^{\circ}$$ being the angle oriented around the vertex. This is because the polygon has been divided into five triangles and $\frac{360}{5}= 72$ .

Varsity pentagon

To solve for the apothem, we can use basic trigonometric ratios:

$tan(54$^{\circ}$)= 1.3764$

$\\1.3764=\frac{x}{2.5\;cm}\\x=3.44\;cm$

Now that we know the apothem length, we can plug in all our values to solve for area:

$Area=5\cdot\frac{1}{2}\cdot 5\;cm \cdot3.44\;cm=43\;cm$

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Example Question #21 : How To Find The Area Of A Pentagon

Each side of this regular pentagon has a length of $7.5\;cm$ . Solve for the area of the pentagon. Round to the nearest tenth.

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Possible Answers:

$37.5\;cm^2$

$96.8\;cm^2$

$35.5\;cm^2$

$48.4\;cm^2$

$140.6\;cm^2$

Correct answer:

$96.8\;cm^2$

Explanation:

When given the value of one side of a regular pentagon, we can assume all sides to be of equal length and we can use this formula to calculate area:

$Area=\frac{s^2 N}{4tan(\frac{180}{N})}$

For thi formula, represents the length of one side while represents the number of sides. Therefore, we would plug in the values as such:

$Area=\frac{(7.5\;cm)^2 \cdot5}{4tan(\frac{180}{5})}=\frac{56.25\;cm^2\cdot 5}{4(0.7265)}=\frac{281.25\;cm^2}{2.906}\approx96.8\;cm^2$

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Example Question #27 : How To Find The Area Of A Pentagon

A regular pentagon has a side length of $17\;cm$ . Find the area rounded to the nearest tenth.

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Possible Answers:

$383.6\;cm^2$

$99.4\;cm^2$

$29.2\;cm^2$

$497.2\;cm^2$

$225.7\;cm^2$

Correct answer:

$497.2\;cm^2$

Explanation:

We can use the following equation to solve for the area of a regular polygon with representing side length and representing number of sides:

$Area=\frac{s^2N}{4tan\frac{180^{\circ}}{N}}$

$\\Area=\frac{(17\;cm)^2\cdot5}{4tan\frac{180^{\circ}}{5}}\\Area=\frac{1,445\;cm^2}{4tan(36^{\circ})}$

$Area=\frac{1,445\;cm^2}{2.9062}\approx497.2\;cm^2$

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Example Question #28 : How To Find The Area Of A Pentagon

In the figure below, find the area of the pentagon if the length of is one-fourth the length of .

Possible Answers:

130

190

230

150

Correct answer:

230

Explanation:

Start by figuring out the lengths of and .

Let be the length of . From the question, we then write the length of as .

From the figure, you should see that AF+FD=AD=BC .

Plug in the variables and solve for .

x+4x=20

5x=20

x=4

So then the length of must be , and the length of must be .

Now, to find the area of the pentagon, notice that we can break the shape down to one rectangle and two right triangles.

Next, find the area of rectangle ABCD .

$\text{Area of rectangle}=\text{length}\times\text{width}$

$\text{Area of ABCD}=9\times 20=180$

Next, find the area of the right triangles.

$\text{Area of Triangle}=\frac{1}{2}(\text{base})(\text{height})$

For triangle AFE ,

$\text{Area}=\frac{1}{2}(4)(5)=10$

For triangle EFD ,

$\text{Area}=\frac{1}{2}(16)(5)=40$

Add up the component areas to find the area of the pentagon.

$\text{Area of Pentagon}=180+10+40=230$

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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

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

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

Example Question #21 : How To Find The Area Of A Pentagon

Example Question #21 : How To Find The Area Of A Pentagon

Example Question #801 : Plane Geometry

Example Question #22 : How To Find The Area Of A Pentagon

Example Question #25 : How To Find The Area Of A Pentagon

Example Question #21 : How To Find The Area Of A Pentagon

Example Question #27 : How To Find The Area Of A Pentagon

Example Question #28 : How To Find The Area Of A Pentagon

All Intermediate Geometry Resources

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