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

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Question

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

Answer

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.

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Question

A regular pentagon has perimeter one yard. Which is the greater quantity?

(A) The length of one side

(B) 7 inches

Answer

One yard is equal to 36 inches. A regular pentagon has five sides of equal length, so one side of the pentagon has length

$36 \div 5 = 7 $\frac{1}{5}$$ inches.

Since $7 $\frac{1}{5}$ > 7$ , (A) is greater.

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Question

A regular pentagon has an area of $\small 32.5$ square units and an apothem measurement of $\small \small 3$ . Find the length of one side of the pentagon.

Answer

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:

$\small \small a=32.5$

So, area of one of the five interior triangles is equal to: $\small \small 32.5\div5=6.5$

Now, apply the area formula:

$\small \small 6.5=\frac{base\times 3}{2}$$

$\small \small 13=base\times3$

$\small \small base=13\div3=4.3$

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Question

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.

Answer

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:

$\small \small a=156$

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}$$

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

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

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Question

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

Answer

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$

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Question

A regular pentagon has a perimeter of $\small \small 75$ . Find the length of one side of the pentagon.

Answer

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.

$\small \small 75=5(s)$

$\small \small s=\frac{75}{5}$=15$

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Question

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.

Answer

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}$$

$\small 32=base\times4.7$

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

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Question

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 $\small x.$

Answer

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:

$\small a=width\times length$

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

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

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Question

A regular pentagon has a perimeter of $\small \small 225$ . Find the length of one side of the pentagon.

Answer

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.

$\small \small 225=5(s)$

$\small \small s=\frac{225}{5}$=45$

Check:

$\small 45\times5=225$

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Question

A regular pentagon has an area of $\small \small 180$ square units and an apothem measurement of $\small \small 7$ . Find the length of one side of the pentagon.

Answer

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:

$\small \small a=180$

So, area of one of the five interior triangles is equal to: $\small \small 180\div5=36$

Now, apply the area formula:

$\small \small 36=\frac{base\times 7}{2}$$

$\small \small 72=base\times7$

$\small \small base=72\div 7=10.3$

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Question

A regular pentagon has a perimeter of $\small \small \small 64$ . Find the length of one side of the pentagon.

Answer

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.

$\small \small 64=5(s)$

$\small \small s=\frac{64}{5}$=12.8$

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Question

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 $\small x.$

Answer

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:

$\small a=width\times length$

$\small \small \small 49.5=9\times x$

$\small \small x=49.5\div9=5.5$

CHECK:

$\small 9\times5.5=49.5$

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Question

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

Answer

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$

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Question

A regular pentagon has an area of $\small \small 365.6$ square units and an apothem measurement of $\small \small 10$ . Find the length of one side of the pentagon.

Answer

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:

$\small \small a=365.6$

So, area of one of the five interior triangles is equal to: $\small \small 365.6\div5=73.12$

Now, apply the area formula:

$\small \small 73.12=\frac{base\times 10}{2}$$

$\small \small 146.24=base\times10$

$\small \small base=146.24\div 10=14.6$

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Question

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

Answer

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$

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Question

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

Answer

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$

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Question

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?

Answer

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.

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Question

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?

Answer

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}$=\frac{2(51)}{5(2.1)}$=9.71$

Make sure to round to two places after the decimal.

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Question

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?

Answer

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}$=\frac{2(200)}{5(7.1)}$=11.27$

Make sure to round to two places after the decimal.

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Question

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?

Answer

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}$=\frac{2(291)}{5(9.1)}$=12.79$

Make sure to round to two places after the decimal.

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