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SSAT Upper Level Math : How to find the area of a pentagon

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Example Questions

Example Question #754 : Geometry

Find the exact area of a pentagon if the side length is $\displaystyle 6$.

Possible Answers:

$\sqrt{25+10\sqrt5}$ $\displaystyle \sqrt{25+10\sqrt5}$

$9\sqrt{10\sqrt5}$ $\displaystyle 9\sqrt{10\sqrt5}$

$15+90\sqrt5$ $\displaystyle 15+90\sqrt5$

$6\sqrt{35\sqrt5}$ $\displaystyle 6\sqrt{35\sqrt5}$

$9\sqrt{25+10\sqrt5}$ $\displaystyle 9\sqrt{25+10\sqrt5}$

Correct answer:

$9\sqrt{25+10\sqrt5}$ $\displaystyle 9\sqrt{25+10\sqrt5}$

Explanation:

Write the formula for the area of a pentagon.

$A=\frac{s^{2}}{4}\sqrt{5(5+2\sqrt5)}$ $\displaystyle A=\frac{s^{2}}{4}\sqrt{5(5+2\sqrt5)}$

Substitute the side length and simplify.

$A=\frac{6^{2}}{4}\sqrt{5(5+2\sqrt5)}$ $\displaystyle A=\frac{6^{2}}{4}\sqrt{5(5+2\sqrt5)}$

$A= 9\sqrt{25+10\sqrt5}$ $\displaystyle A= 9\sqrt{25+10\sqrt5}$

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Example Question #755 : Geometry

In terms of $\displaystyle x$, find the area of a regular pentagon that has a side length of $\displaystyle 6x$ and an apothem of $\displaystyle 2x$.

Possible Answers:

30x^2 $\displaystyle 30x^2$

30x $\displaystyle 30x$

60x^2 $\displaystyle 60x^2$

120x^2 $\displaystyle 120x^2$

Correct answer:

30x^2 $\displaystyle 30x^2$

Explanation:

To find the area of a regular polygon,

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

To find the perimeter of the pentagon,

$\text{Perimeter}=5(side)$ $\displaystyle \text{Perimeter}=5(side)$

For the given pentagon,

$\text{Perimeter}=5\times 6x=30x$ $\displaystyle \text{Perimeter}=5\times 6x=30x$

So then, to find the area of the pentagon,

$\text{Area}=\frac{1}{2}(30x)(2x)=30x^2$ $\displaystyle \text{Area}=\frac{1}{2}(30x)(2x)=30x^2$

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Example Question #756 : Geometry

Find the area of a regular pentagon that has a side length of $\displaystyle 20$ and an apothem of $\displaystyle 15$.

Possible Answers:

1500 $\displaystyle 1500$

500 $\displaystyle 500$

2000 $\displaystyle 2000$

750 $\displaystyle 750$

Correct answer:

750 $\displaystyle 750$

Explanation:

To find the area of a regular polygon,

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

To find the perimeter of the pentagon,

$\text{Perimeter}=5(side)$ $\displaystyle \text{Perimeter}=5(side)$

For the given pentagon,

$\text{Perimeter}=5\times20=100$ $\displaystyle \text{Perimeter}=5\times20=100$

So then, to find the area of the pentagon,

$\text{Area}=\frac{1}{2}(100)(15)=750$ $\displaystyle \text{Area}=\frac{1}{2}(100)(15)=750$

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Example Question #21 : Areas And Perimeters Of Polygons

Find the area of a regular pentagon with a side length of $\displaystyle 4$.

Possible Answers:

41.21 $\displaystyle 41.21$

98.21 $\displaystyle 98.21$

29.11 $\displaystyle 29.11$

27.53 $\displaystyle 27.53$

Correct answer:

27.53 $\displaystyle 27.53$

Explanation:

Use the following formula to find the area of a regular pentagon:

$\text{Area}=\frac{1}{4}\sqrt{5(5+2\sqrt5)}(side)^2$ $\displaystyle \text{Area}=\frac{1}{4}\sqrt{5(5+2\sqrt5)}(side)^2$

Plugging in the information given by the question,

$\text{Area}=\frac{1}{4}\sqrt{5(5+2\sqrt5)}(4)^2=27.53$ $\displaystyle \text{Area}=\frac{1}{4}\sqrt{5(5+2\sqrt5)}(4)^2=27.53$

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Example Question #758 : Geometry

In terms of $\displaystyle z$, find the area of a regular pentagon with side lengths of $\displaystyle 5z$ and an apothem of $\displaystyle 6$.

Possible Answers:

75z $\displaystyle 75z$

50z $\displaystyle 50z$

150z $\displaystyle 150z$

75z^2 $\displaystyle 75z^2$

Correct answer:

75z $\displaystyle 75z$

Explanation:

To find the area of a regular polygon,

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

To find the perimeter of the pentagon,

$\text{Perimeter}=5(side)$ $\displaystyle \text{Perimeter}=5(side)$

For the given pentagon,

$\text{Perimeter}=5\times 5z =25z$ $\displaystyle \text{Perimeter}=5\times 5z =25z$

So then, to find the area of the pentagon,

$\text{Area}=\frac{1}{2}(25z)(6)=75z$ $\displaystyle \text{Area}=\frac{1}{2}(25z)(6)=75z$

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Example Question #761 : Geometry

In terms of $\displaystyle a$, find the area of a regular pentagon that has a side length of $\displaystyle 8a$ and an apothem of $\displaystyle 4a$.

Possible Answers:

160a^2 $\displaystyle 160a^2$

120a^2 $\displaystyle 120a^2$

80a $\displaystyle 80a$

80a^2 $\displaystyle 80a^2$

Correct answer:

80a^2 $\displaystyle 80a^2$

Explanation:

To find the area of a regular polygon,

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

To find the perimeter of the pentagon,

$\text{Perimeter}=5(side)$ $\displaystyle \text{Perimeter}=5(side)$

For the given pentagon,

$\text{Perimeter}=5\times 8a=40a$ $\displaystyle \text{Perimeter}=5\times 8a=40a$

So then, to find the area of the pentagon,

$\text{Area}=\frac{1}{2}(40a)(4a)=80a^2$ $\displaystyle \text{Area}=\frac{1}{2}(40a)(4a)=80a^2$

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Example Question #762 : Geometry

In terms of $\displaystyle b$, find the area of a regular pentagon with side lengths of 12b $\displaystyle 12b$ and an apothem of 10b $\displaystyle 10b$.

Possible Answers:

200b^2 $\displaystyle 200b^2$

300b^2 $\displaystyle 300b^2$

600b^2 $\displaystyle 600b^2$

400b^2 $\displaystyle 400b^2$

Correct answer:

300b^2 $\displaystyle 300b^2$

Explanation:

To find the area of a regular polygon,

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

To find the perimeter of the pentagon,

$\text{Perimeter}=5(side)$ $\displaystyle \text{Perimeter}=5(side)$

For the given pentagon,

$\text{Perimeter}=5\times 12b=60b$ $\displaystyle \text{Perimeter}=5\times 12b=60b$

So then, to find the area of the pentagon,

$\text{Area}=\frac{1}{2}(60b)(10b)=300b^2$ $\displaystyle \text{Area}=\frac{1}{2}(60b)(10b)=300b^2$

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Example Question #763 : Geometry

The area of a regular polygon is 512 $\displaystyle 512$. If the length of its apothem is $\displaystyle 8$, what is the length of each side of the pentagon?

Possible Answers:

22.2 $\displaystyle 22.2$

18.2 $\displaystyle 18.2$

25.6 $\displaystyle 25.6$

20.8 $\displaystyle 20.8$

Correct answer:

25.6 $\displaystyle 25.6$

Explanation:

To find the area of a regular polygon,

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

For the given pentagon,

$512=\frac{1}{2}(Perimeter)(8)$ $\displaystyle 512=\frac{1}{2}(Perimeter)(8)$

$\displaystyle 8(Perimeter)=1024$

$\displaystyle Perimeter=128$

To find the length of each side of the pentagon, divide the perimeter by $\displaystyle 5$.

$\displaystyle Side=25.6$

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Example Question #21 : Area Of Polygons

A regular pentagon has an area of 250 $\displaystyle 250$. Find the length of each side of the pentagon if the apothem has a length of $\displaystyle 5$.

Possible Answers:

$\displaystyle 10$

$\displaystyle 15$

$\displaystyle 20$

$\displaystyle 25$

Correct answer:

$\displaystyle 20$

Explanation:

To find the area of a regular polygon,

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

For the given pentagon,

$250=\frac{1}{2}(Perimeter)(5)$ $\displaystyle 250=\frac{1}{2}(Perimeter)(5)$

$\displaystyle 5(Perimeter)=500$

$\displaystyle Perimeter=100$

To find the length of each side of the pentagon, divide the perimeter by $\displaystyle 5$.

Side=20 $\displaystyle Side=20$

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Example Question #22 : Area Of Polygons

The area of a regular pentagon is 410 $\displaystyle 410$. Find the length of a side of the pentagon if the apothem has a length of $\displaystyle 12$.

Possible Answers:

$\frac{41}{3}$ $\displaystyle \frac{41}{3}$

$\frac{61}{3}$ $\displaystyle \frac{61}{3}$

$\displaystyle 20$

$\frac{15}{2}$ $\displaystyle \frac{15}{2}$

Correct answer:

$\frac{41}{3}$ $\displaystyle \frac{41}{3}$

Explanation:

To find the area of a regular polygon,

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

For the given pentagon,

$410=\frac{1}{2}(Perimeter)(12)$ $\displaystyle 410=\frac{1}{2}(Perimeter)(12)$

$\displaystyle 12(Perimeter)=820$

$Perimeter=\frac{820}{12}$ $\displaystyle Perimeter=\frac{820}{12}$

To find the length of each side of the pentagon, divide the perimeter by $\displaystyle 5$.

$Side=\frac{820}{60}=\frac{41}{3}$ $\displaystyle Side=\frac{820}{60}=\frac{41}{3}$

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SSAT Upper Level Math : How to find the area of a pentagon

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #754 : Geometry

Example Question #755 : Geometry

Example Question #756 : Geometry

Example Question #21 : Areas And Perimeters Of Polygons

Example Question #758 : Geometry

Example Question #761 : Geometry

Example Question #762 : Geometry

Example Question #763 : Geometry

Example Question #21 : Area Of Polygons

Example Question #22 : Area Of Polygons

All SSAT Upper Level Math Resources

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