How to find the length of the side of a pentagon

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Math › How to find the length of the side of a pentagon

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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 pentagon with a perimeter of one mile has three congruent sides. The second-longest side is 250 feet longer than any one of those three congruent sides, and the longest side is 500 feet longer than the second-longest side.

Which is the greater quantity?

(a) The length of the longest side of the pentagon

(b) Twice the length of one of the three shortest sides of the pentagon

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Explanation

If each of the five congruent sides has measure , then the other two sides have measures and $\left (x + 250 \right ) +500 = x + 750$ . Add the sides to get the perimeter, which is equal to feet, the solve for :

$5x\div 5 = 4,280 \div 5$

feet

Now we can compare (a) and (b).

(a) The longest side has measure feet.

(b) The three shortest sides each have length 856 feet; twice this is $856 \times 2 = 1,712$ feet.

(b) is greater.

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.

Find the length of the side of the following pentagon.

Angle_length_of_side_pentagon

The perimeter of the pentagon is $20\ m$ .

$4\ m$

$5\ m$

$7\ m$

$3\ m$

$9\ m$

Explanation

The formula for the perimeter of a regular pentagon is

where represents the length of the side.

Plugging in our values, we get:

$20\ m = 5(s)$

$s=4\ m$

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

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 $\small 5.$