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Intermediate Geometry : How to find the length of the side of a pentagon

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Example Questions

Example Question #21 : Pentagons

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

Possible Answers:

$\small 33$

$\small 24.6$

$\small 27.4$

$\small 26.6$

$\small 24.4$

Correct answer:

$\small 26.6$

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.

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

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

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Example Question #22 : Pentagons

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.

Possible Answers:

$\small 36.5$

$\small 16.25$

$\small 14.6$

$\small 12.5$

$\small 16.5$

Correct answer:

$\small 14.6$

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:

$\small \small a=365.6$

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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Example Question #31 : Pentagons

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

Possible Answers:

Correct answer:

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.

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

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Example Question #771 : Intermediate Geometry

If a regular pentagon has an area of and an apothem of 1.2 , what is the length of a side of the pentagon?

Possible Answers:

Correct answer:

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.

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

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Example Question #772 : Intermediate Geometry

If the area of a regular pentagon is 20.5 and the length of the apothem is 1.9 , what is the length of a side of the pentagon?

Possible Answers:

6.77

5.19

3.34

4.32

Correct answer:

4.32

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.

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Example Question #773 : Intermediate Geometry

If the area of a regular pentagon is 200 and the length of the apothem is 7.1 , what is the length of a side of the pentagon?

Possible Answers:

13.98

11.27

10.55

9.52

Correct answer:

11.27

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.

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

Make sure to round to two places after the decimal.

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Example Question #774 : Intermediate Geometry

If the area of a regular pentagon is and the length of the apothem is 2.1 , what is the length of a side of the pentagon?

Possible Answers:

9.71

10.01

7.44

8.45

Correct answer:

9.71

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.

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

Make sure to round to two places after the decimal.

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Example Question #775 : Intermediate Geometry

If the area of a regular pentagon is 291 and the length of the apothem is 9.1 , what is the length of a side of the pentagon?

Possible Answers:

14.29

12.79

11.12

13.01

Correct answer:

12.79

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.

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

Make sure to round to two places after the decimal.

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Example Question #776 : Intermediate Geometry

If the area of a regular pentagon is 105 and the length of the apothem is 4.1 , what is the length of a side of the pentagon?

Possible Answers:

11.95

9.50

12.41

10.24

Correct answer:

10.24

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.

$\text{side}=\frac{2(105)}{5(4.1)}=10.24$

Make sure to round to two places after the decimal.

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Example Question #777 : Intermediate Geometry

If the area of a regular pentagon is 155 and the length of the apothem is 6.2 , what is the length of a side of the pentagon?

Possible Answers:

12.50

9.25

8.75

Correct answer:

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.

$\text{side}=\frac{2(155)}{5(6.2)}=10$

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Intermediate Geometry : How to find the length of the side of a pentagon

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #21 : Pentagons

Example Question #22 : Pentagons

Example Question #31 : Pentagons

Example Question #771 : Intermediate Geometry

Example Question #772 : Intermediate Geometry

Example Question #773 : Intermediate Geometry

Example Question #774 : Intermediate Geometry

Example Question #775 : Intermediate Geometry

Example Question #776 : Intermediate Geometry

Example Question #777 : Intermediate Geometry

All Intermediate Geometry Resources

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