How to find the area of a pentagon - SSAT Upper Level Quantitative (Math)

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Question

Find the exact area of a pentagon if the side length is .

Answer

Write the formula for the area of a pentagon.

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

Substitute the side length and simplify.

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

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

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Question

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

Answer

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,

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

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Question

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

Answer

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,

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

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Question

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

Answer

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,

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

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Question

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

Answer

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,

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

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Question

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

Answer

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,

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

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Question

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

Answer

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,

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

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Question

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

Answer

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,

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

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Question

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

Answer

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,

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

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Question

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

Answer

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,

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

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Question

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

Answer

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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Question

A regular pentagon has an area of . Find the length of each side of the pentagon if the apothem has a length of .

Answer

To find the area of a regular polygon,

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

For the given pentagon,

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

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

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Question

The area of a regular pentagon is . Find the length of a side of the pentagon if the apothem has a length of .

Answer

To find the area of a regular polygon,

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

For the given pentagon,

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

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

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

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

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