Geometry - How to find the length of the side of a pentagon

If a regular pentagon has an area of and an apothem length of , what is the length of a side of the pentagon?

Explanation

Recall how to find the area of a regular pentagon:

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

Now, the perimeter of a regular pentagon can be found by multiplying the side length by :

$\text{perimeter}=5(\text{side})$

Substitute this into the equation for the area.

$\text{Area}=\frac{1}{2}(\text{apothem})(5(\text{side}))$

Now, rearrange the equation to solve for the side length.

$5(\text{apothem})(\text{side})=2(\text{Area})$

$\text{side}=\frac{2(\text{Area})}{5(\text{apothem})}$

Plug in the given area and apothem to solve for the side length of the pentagon.

$\text{side}=\frac{2(50)}{5(10)}=2$

A regular pentagon has a perimeter of $\small \small 133$ . Find the length of one side of the pentagon.

$\small 26.6$

$\small 24.4$

$\small 24.6$

$\small 33$

$\small 27.4$

Explanation

A regular pentagon must have five equivalent sides and five equivalent interior angles.

This problem provides the measurement for the total perimeter of the pentagon, work backwards using the formula:

$\small p=5(s)$ , where $\small s =$ the length of one side of the pentagon.

$\small \small 133=5(s)$

$\small \small s=\frac{133}{5}=26.6$

If a regular pentagon has an area of and an apothem of , what is the length of a side of the pentagon?

Explanation

Recall how to find the area of a regular pentagon:

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

Now, the perimeter of a regular pentagon can be found by multiplying the side length by :

$\text{perimeter}=5(\text{side})$

Substitute this into the equation for the area.

$\text{Area}=\frac{1}{2}(\text{apothem})(5(\text{side}))$

Now, rearrange the equation to solve for the side length.

$5(\text{apothem})(\text{side})=2(\text{Area})$

$\text{side}=\frac{2(\text{Area})}{5(\text{apothem})}$

Plug in the given area and apothem to solve for the side length of the pentagon.

$\text{side}=\frac{2(12)}{5(1.2)}=4$

In pentagon , if the length of is one-fifth the length of , what is the length of the side ?

Explanation

Notice that this pentagon is made up of one rectangle and two right triangles. In order to find the length of , we will first need to find the length of .

Let be the length of , so then the length of can be represented by .

Since we have rectangle , we know that .

Thus,

Now, plug in the variables and solve for .

Thus, we know that $AF=10\text{ and }FD=50$ .

Now, use the Pythagorean theorem to find the length of .

$ED=\sqrt{(EF)^2+(FD)^2}$

$ED=\sqrt{12^2+50^2}=51.42$

Make sure to round to places after the decimal.

A regular pentagon has an area of $\small 80$ square units and an apothem measurement of $\small 4.7$ . Find the length of one side of the pentagon.

$\small 6.8$

$\small 8.6$

$\small 4.3$

$\small 4.2$

$\small 16$

Explanation

A regular pentagon must have five equivalent sides and five equivalent interior angles. Regular pentagons can be divided up into five equivalent interior triangles, where the base of the triangle is a side length and the height of the triangle is the apothem. This problem provides the total area for the pentagon.

Therefore, you must work backwards using the area formula: $\small a=\frac{base\times height}{2}$ , where the base is equal to the length of one side of the triangle and the height of the triangle is the measurement of the apothem. However, this will only provide the measurement for one of the five interior triangles, thus you first need to divide the total area by $\small 5.$

The solution is:

$\small a=80$

So, area of one of the five interior triangles is equal to: $\small 80\div5=16$

Now, apply the area formula:

$\small 16=\frac{base\times 4.7}{2}$

$\small 32=base\times4.7$

$\small base=32\div4.7=6.8$

If the area of a regular pentagon is , and the length of the apothem is , what is the length of a side of the pentagon?

Explanation

Recall how to find the area of a regular pentagon:

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

Now, the perimeter of a regular pentagon can be found by multiplying the side length by :

$\text{perimeter}=5(\text{side})$

Substitute this into the equation for the area.

$\text{Area}=\frac{1}{2}(\text{apothem})(5(\text{side}))$

Now, rearrange the equation to solve for the side length.

$5(\text{apothem})(\text{side})=2(\text{Area})$

$\text{side}=\frac{2(\text{Area})}{5(\text{apothem})}$

Plug in the given area and apothem to solve for the side length of the pentagon.

$\text{side}=\frac{2(561)}{5(13.4)}=16.75$

Make sure to round to two places after the decimal.

A regular pentagon has a perimeter of $\small 125$ . Find the length of one side of the pentagon.

$\small 25$

$\small 75$

$\small 15$

$\small 20$

$\small 35$

Explanation

By definition a regular pentagon must have five equivalent sides and five equivalent interior angles. Since this problem provides the measurement for the total perimeter of the pentagon, work backwards using the formula:

$\small p=5(s)$ , where $\small s =$ the length of one side of the pentagon.

$\small 125=5(s)$

$\small s=\frac{125}{5}=25$

A regular pentagon has an area of $\small \small 156$ square units and an apothem measurement of $\small \small 6.6$ . Find the length of one side of the pentagon.

$\small \small 9.5$

$\small 6.6$

$\small 6.9$

$\small 7.25$

$\small 9.9$

Explanation

The solution is:

$\small \small a=156$

So, area of one of the five interior triangles is equal to: $\small \small 156\div5=31.2$

Now, apply the area formula:

$\small \small 31.2=\frac{base\times 6.6}{2}$

$\small \small 62.4=base\times6.6$

$\small \small base=62.4\div 6.6= 9.5$

If the area of a regular pentagon is and the length of the apothem is , what is the length of a side of the pentagon?

Explanation

Recall how to find the area of a regular pentagon:

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

Now, the perimeter of a regular pentagon can be found by multiplying the side length by :

$\text{perimeter}=5(\text{side})$

Substitute this into the equation for the area.

$\text{Area}=\frac{1}{2}(\text{apothem})(5(\text{side}))$

Now, rearrange the equation to solve for the side length.

$5(\text{apothem})(\text{side})=2(\text{Area})$

$\text{side}=\frac{2(\text{Area})}{5(\text{apothem})}$

Plug in the given area and apothem to solve for the side length of the pentagon.

$\text{side}=\frac{2(20.5)}{5(1.9)}=4.32$

Make sure to round to two places after the decimal.

Pentagon_series_vt_custom

The pentagon shown above has been divided into a top portion with two equivalent right triangles and a bottom rectangular portion. Find the length of side $\small x.$

$\small 12$

$\small 12.5$

$\small 8.5$

$\small 8.25$

$\small 6.25$

Explanation

To solve this problem notice that the area is provided for the bottom rectangular portion of the pentagon. Therefore, work backwards using the area formula:

$\small a=width\times length$

$\small \small 174=14.5\times x$

$\small x=174\div14.5=12$

How to find the length of the side of a pentagon

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