How to find the area of a pentagon - Intermediate Geometry

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Question

A regular pentagon has a side length of inches and an apothem length of inches. Find the area of the pentagon.

Answer

By definition a regular pentagon must have equal sides and equivalent interior angles. Since we are told that this pentagon has a side length of inches, all of the sides must have a length of inches. Additionally, the question provides the length of the apothem of the pentagon--which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

$A=\frac{base\times height}{2}$

$A=\frac{9\times 6.2}{2}=\frac{55.8}{2}=27.9$

Note: the area shown above is only the a measurement from one of the five total interior triangles. Thus, to find the total area of the pentagon multiply:

$27.9\times5=139.5$

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Question

A regular pentagon has a side length of and an apothem length of . Find the area of the pentagon.

Answer

By definition a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon--which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

$A=\frac{base\times height}{2}$

$A=\frac{14\times 9.6}{2}=\frac{134.4}{2}=67.2$

Note: is only the measurement for one of the five interior triangles. Thus, the final solution is:

$67.2\times5=336$

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Question

Find the area of the pentagon shown above.

Answer

To find the area of this pentagon, divide the interior of the pentagon into a four-sided rectangle and two right triangles. The area of the bottom rectangle can be found using the formula:

$A=width\times length$

$A=20\times15=300$

The area of the two right triangles can be found using the formula:

$A=\frac{base\times height}{2}$

$A=\frac{10\times 8}{2}=\frac{80}{2}=40$

Since there are two right triangles, the sum of both will equal the area of the entire triangular top portion of the pentagon.

Thus, the solution is:

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Question

A regular pentagon has a perimeter of yards and an apothem length of yards. Find the area of the pentagon.

Answer

To solve this problem, first work backwards using the perimeter formula for a regular pentagon:

$s=\frac{60}{5}=12$

Now you have enough information to find the area of this regular triangle.
Note: a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon—which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

$A=\frac{base\times height}{2}$

$A=\frac{8.3\times 12}{2}=\frac{99.6}{2}=49.8$

Thus, the area of the entire pentagon is:

$A=49.8\times 5=249$

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Question

A regular pentagon has a side length of and an apothem length of . Find the area of the pentagon.

Answer

By definition a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon—which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

$A=\frac{base\times height}{2}$

$A=\frac{15\times 10.3}{2}=\frac{154.5}{2}=77.25$

Keep in mind that this is the area for only one of the five total interior triangles.

The total area of the pentagon is:

$A=77.25\times 5=386.25$

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Question

A regular pentagon has a perimeter of and an apothem length of . Find the area of the pentagon.

Answer

To solve this problem, first work backwards using the perimeter formula for a regular pentagon:

$s=\frac{50}{5}=10$

Now you have enough information to find the area of this regular triangle.
Note: a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon—which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

$A=\frac{base\times height}{2}$

$A=\frac{10\times 7}{2}=\frac{70}{2}=35$

To find the total area of the pentagon multiply:

$35\times5=175$

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Question

A regular pentagon has a side length of and an apothem length of . Find the area of the pentagon.

Answer

By definition a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon—which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

$A=\frac{base\times height}{2}$

$A=\frac{25\times 17.2}{2}=\frac{430}{2}=215$

However, the total area of the pentagon is equal to:

$215\times 5= 1,075$

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Question

Find the area of the pentagon shown above.

Answer

To find the area of this pentagon, divide the interior of the pentagon into a four-sided rectangle and two right triangles. The area of the bottom rectangle can be found using the formula:

$A=width\times length$

$A=26\times16=416$

The area of the two right triangles can be found using the formula:

$A=\frac{base\times height}{2}$

$A=\frac{13\times 12}{2}=\frac{156}{2}=78$

Since there are two right triangles, the sum of both will equal the area of the entire triangular top portion of the pentagon.

Thus, the solution is:

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Question

A regular pentagon has a side length of and an apothem length of . Find the area of the pentagon.

Answer

By definition a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon—which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

$A=\frac{base\times height}{2}$

$A=\frac{7\times 4.8}{2}=\frac{33.6}{2}=16.8$

Note: is only the measurement for one of the five interior triangles. Thus, the solution is:

$16.8\times5=84$

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Question

A regular pentagon has a perimeter of and an apothem length of . Find the area of the pentagon.

Answer

To solve this problem, first work backwards using the perimeter formula for a regular pentagon:

$s=\frac{65}{5}=13$

Now you have enough information to find the area of this regular triangle.
Note: a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon—which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

$A=\frac{base\times height}{2}$

$A=\frac{8.9\times 13}{2}=\frac{115.7}{2}=57.85$

Thus, the area of the entire pentagon is:

$A=57.85\times 5=289.25$

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Question

Find the area of the pentagon shown above.

Answer

To find the area of this pentagon, divide the interior of the pentagon into a four-sided rectangle and two right triangles. The area of the bottom rectangle can be found using the formula:

$A=width\times length$

$A=18\times8=144$

The area of the two right triangles can be found using the formula:

$A=\frac{base\times height}{2}$

$A=\frac{9\times 6}{2}=\frac{54}{2}=27$

Since there are two right triangles, the sum of both will equal the area of the entire triangular top portion of the pentagon.

Thus, the solution is:

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Question

A regular pentagon has a perimeter of and an apothem length of . Find the area of the pentagon.

Answer

To solve this problem, first work backwards using the perimeter formula for a regular pentagon:

$s=\frac{145}{5}=29$

Now you have enough information to find the area of this regular triangle.
Note: a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon—which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

$A=\frac{base\times height}{2}$

$A=\frac{29\times 20}{2}=\frac{580}{2}=290$

Thus, the area of the entire pentagon is:

$A=290\times 5=1,450$

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Question

Each side of this pentagon has a length of .

Solve for the area of the pentagon.

Answer

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:

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 $\frac{360}{5}= 72$ .

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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Question

Each side of this regular pentagon has a length of . Solve for the area of the pentagon. Round to the nearest tenth.

Answer

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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Question

A regular pentagon has a side length of . Find the area rounded to the nearest tenth.

Answer

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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Question

Find the area of the regular pentagon.

Answer

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.

$\text{Perimeter}=5(3)$

$\text{Perimeter}=15$

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

$\text{Area of Pentagon}=\frac{1}{2}(2)(15)$

$\text{Area of Pentagon}=15$

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Question

Find the area of the regular pentagon.

Answer

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.

$\text{Perimeter}=5(14)$

$\text{Perimeter}=70$

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

$\text{Area of Pentagon}=\frac{1}{2}(12)(70)$

$\text{Area of Pentagon}=420$

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Question

Find the area of the regular pentagon.

Answer

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.

$\text{Perimeter}=5(8)$

$\text{Perimeter}=40$

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

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

$\text{Area of Pentagon}=60$

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Question

Find the area of the regular pentagon.

Answer

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.

$\text{Perimeter}=5(7)$

$\text{Perimeter}=35$

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

$\text{Area of Pentagon}=\frac{1}{2}(4)(35)$

$\text{Area of Pentagon}=70$

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Question

Find the area of the regular pentagon.

Answer

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.

$\text{Perimeter}=5(16)$

$\text{Perimeter}=80$

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

$\text{Area of Pentagon}=\frac{1}{2}(14)(80)$

$\text{Area of Pentagon}=560$

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