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

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Question

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

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Answer

Write the formula for the area of a pentagon.

Substitute the side length and simplify.

$A= 9$\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 .

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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 .

Tap to reveal answer

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 .

Tap to reveal answer

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 .

Tap to reveal answer

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,

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Question

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

Tap to reveal answer

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 .

Tap to reveal answer

Answer

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

Plugging in the information given by the question,

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Question

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

Tap to reveal answer

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 .

Tap to reveal answer

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,

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Question

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

Tap to reveal answer

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,

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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?

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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 .

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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 .

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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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Question

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

Tap to reveal answer

Answer

Write the formula for the area of a pentagon.

Substitute the side length and simplify.

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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$

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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$

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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$

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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,

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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 .

Tap to reveal answer

Answer

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

Plugging in the information given by the question,

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