How to find the area of a pentagon

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SSAT Upper Level Quantitative › How to find the area of a pentagon

Questions 1 - 10

1

Find the area of a regular pentagon that has a side length of and an apothem of .

Explanation

To find the area of a regular polygon,

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

To find the perimeter of the pentagon,

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

For the given pentagon,

$\text{Perimeter}=5\times6=30$

So then, to find the area of the pentagon,

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

2

Find the area of a regular pentagon with a side length of and an apothem of .

Explanation

To find the area of a regular polygon,

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

To find the perimeter of the pentagon,

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

For the given pentagon,

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

So then, to find the area of the pentagon,

$\text{Area}=\frac{1}{2}(60)(4)=120$

3

Find the area of a regular pentagon that has a side length of and an apothem of .

Explanation

To find the area of a regular polygon,

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

To find the perimeter of the pentagon,

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

For the given pentagon,

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

So then, to find the area of the pentagon,

$\text{Area}=\frac{1}{2}(60)(3)=90$

4

In terms of , find the area of a regular pentagon that has a side length of and an apothem of .

Explanation

To find the area of a regular polygon,

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

To find the perimeter of the pentagon,

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

For the given pentagon,

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

So then, to find the area of the pentagon,

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

5

Find the area of a regular pentagon that has a side length of and an apothem of .

Explanation

To find the area of a regular polygon,

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

To find the perimeter of the pentagon,

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

For the given pentagon,

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

So then, to find the area of the pentagon,

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

6

Find the area of a regular pentagon with a side length of .

Explanation

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

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

Plugging in the information given by the question,

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

7

In terms of , find the area of a regular pentagon with side lengths of and an apothem of .

Explanation

To find the area of a regular polygon,

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

To find the perimeter of the pentagon,

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

For the given pentagon,

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

So then, to find the area of the pentagon,

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

8

In terms of , find the area of a regular pentagon that has a side length of and an apothem of .

Explanation

To find the area of a regular polygon,

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

To find the perimeter of the pentagon,

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

For the given pentagon,

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

So then, to find the area of the pentagon,

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

9

In terms of , find the area of a regular pentagon with side lengths of and an apothem of .

Explanation

To find the area of a regular polygon,

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

To find the perimeter of the pentagon,

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

For the given pentagon,

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

So then, to find the area of the pentagon,

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

10

The area of a regular polygon is . If the length of its apothem is , what is the length of each side of the pentagon?

Explanation

To find the area of a regular polygon,

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

For the given pentagon,

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

To find the length of each side of the pentagon, divide the perimeter by .

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