Geometry - How to find the area of a pentagon

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

sq. units

Explanation

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:

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

Each side of this pentagon has a length of .

Solve for the area of the pentagon.

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

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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:

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

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

square units

Explanation

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

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:

$A=77.25\times 5=386.25$

Find the area of the regular pentagon.

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.

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

$\text{Area of Pentagon}=1260$

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

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 .

Plug in the variables and solve for .

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 .

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

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

Next, find the area of the right triangles.

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

For triangle ,

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

For triangle ,

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

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

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

Find the area of the regular pentagon.

None of these

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.

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

$\text{Perimeter}=15$

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

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

$\text{Area of Pentagon}=15$

Find the area of the regular pentagon.

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.

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

$\text{Perimeter}=70$

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

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

$\text{Area of Pentagon}=420$

Find the area of the regular pentagon.

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.

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

$\text{Perimeter}=40$

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

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

$\text{Area of Pentagon}=60$

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

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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:

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

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

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

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

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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:

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

How to find the area of a pentagon

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