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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #1 : How To Find The Length Of The Side Of A Pentagon

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.

Possible Answers:

$\small 3.9$

$\small 4.3$

$\small 3.4$

$\small 2.9$

$\small 5.3$

Correct answer:

$\small 4.3$

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:

$\small \small a=32.5$

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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Example Question #2 : How To Find The Length Of The Side Of A Pentagon

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 $\small x.$

Possible Answers:

$\small 8.5$

$\small 6.25$

$\small 12$

$\small 8.25$

$\small 12.5$

Correct answer:

$\small 12$

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:

$\small a=width\times length$

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

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

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Example Question #2 : How To Find The Length Of The Side Of A Pentagon

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

Possible Answers:

$\small 40$

$\small 30$

$\small 35$

$\small 45$

$\small 55$

Correct answer:

$\small 45$

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.

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

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

Check:

$\small 45\times5=225$

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Example Question #2 : How To Find The Length Of The Side Of A Pentagon

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.

Possible Answers:

$\small \small 10.3$

$\small 8.9$

$\small 10.9$

$\small 6.25$

$\small 7.25$

Correct answer:

$\small \small 10.3$

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:

$\small \small a=180$

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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Example Question #1 : How To Find The Length Of The Side Of A Pentagon

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

Possible Answers:

$\small 9.25$

$\small 8.8$

$\small 12.8$

$\small 12$

$\small 10.7$

Correct answer:

$\small 12.8$

Explanation:

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Example Question #1 : How To Find The Length Of The Side Of A Pentagon

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.

Possible Answers:

$\small 6.9$

$\small \small 9.5$

$\small 7.25$

$\small 6.6$

$\small 9.9$

Correct answer:

$\small \small 9.5$

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:

$\small \small a=156$

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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Example Question #1 : How To Find The Length Of The Side Of A Pentagon

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 $\small x.$

Possible Answers:

$\small 8.5$

$\small 7.25$

$\small 6.25$

$\small 6.5$

$\small 5.5$

Correct answer:

$\small 5.5$

Explanation:

Report an Error

Example Question #761 : Intermediate Geometry

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

Possible Answers:

$\small 24.4$

$\small 33$

$\small 26.6$

$\small 24.6$

$\small 27.4$

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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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #1 : How To Find The Length Of The Side Of A Pentagon

Example Question #2 : How To Find The Length Of The Side Of A Pentagon

Example Question #2 : How To Find The Length Of The Side Of A Pentagon

Example Question #2 : How To Find The Length Of The Side Of A Pentagon

Example Question #1 : How To Find The Length Of The Side Of A Pentagon

Example Question #1 : How To Find The Length Of The Side Of A Pentagon

Example Question #1 : How To Find The Length Of The Side Of A Pentagon

Example Question #761 : Intermediate Geometry

Example Question #22 : Pentagons

Example Question #31 : Pentagons

All Intermediate Geometry Resources

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