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Intermediate Geometry : Pentagons

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

If a side length of a pentagon is 2a+b , what is the perimeter of the pentagon?

Possible Answers:

10a

5a+b

10a+5b

10a+b

10a+10b

Correct answer:

10a+5b

Explanation:

Write the formula to find the perimeter of a pentagon.

P=5s

Substitute the side length 2a+b into the equation and simplify.

P=5(2a+b)=10a+5b

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Example Question #3 : How To Find The Perimeter Of A Pentagon

If the side of a pentagon is $5\sqrt5$ , what is the perimeter?

Possible Answers:

$5\sqrt5$

$5+5\sqrt5$

125

$10\sqrt5$

$25\sqrt5$

Correct answer:

$25\sqrt5$

Explanation:

Write the perimeter formula for pentagons and substitute the side length.

P=5s

$P=5(5\sqrt5)=25\sqrt5$

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

Determine the perimeter of a pentagon if the side length is $\frac{\sqrt2}{10}$ .

Possible Answers:

$\frac{\sqrt2}{2}$

$\frac{1}{2}$

$\frac{\sqrt10}{10}$

$\sqrt2$

Correct answer:

$\frac{\sqrt2}{2}$

Explanation:

Write the formula to find the perimeter for a pentagon.

P=5s

Substitute the side into the perimeter formula.

$P=5(\frac{\sqrt2}{10})=\frac{\sqrt2}{2}$

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Example Question #5 : How To Find The Perimeter Of A Pentagon

In the figure below, if the length of is one-third the length of , what is the perimeter of the pentagon ABCDE ?

Possible Answers:

56.99

78.08

65.65

60.33

Correct answer:

65.65

Explanation:

Notice that the pentagon is made up of one rectangle and two right triangles. Start by finding the lengths of and .

Let be the length of . Then, it follows that is the length of .

From the figure, we know that AF+FD=AD . Since we are working with a rectangle, AD=BC=24 .

Now, solve for .

x+3x=24

4x=24

x=6

So then, $AF=6 \text{ and }FD=18$ .

Now that we know AF=6 , we can use the Pythagorean theorem to solve for the length of .

$AE=\sqrt{(AF)^2+(FE)^2}$

Plug in the lengths of the legs to find the length of the hypotenuse.

$AE=\sqrt{6^2+4^2}=\sqrt{52}$

Next, find the length of by using the Pythagorean theorem.

$DE=\sqrt{(FD)^2+(EF)^2}$

$DE=\sqrt{18^2+4^2}=\sqrt{340}$

Finally, add up all the side lengths of the pentagon to find its perimeter.

$\text{Perimeter}=AB+BC+CD+DE+EA$

$\text{Perimeter}=8+24+8+\sqrt{340}+\sqrt{52}=65.65$

Make sure to round to places after the decimal.

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Example Question #6 : How To Find The Perimeter Of A Pentagon

In pentagon ABCDE below, the length of is one-third that of . Find the perimeter of the pentagon.

Possible Answers:

146.41

128.57

133.86

152.09

Correct answer:

146.41

Explanation:

Notice that this pentagon is made up of two right triangles and one rectangle.

Start by finding the length of . Since we are given an angle measurement and a length of a leg of the triangle, we can use tangent to find the length of .

$\tan 38=\frac{9}{AF}$

$AF=\frac{9}{\tan38}=11.52$

Now, we know that is three times the length of .

FD=3(AF)=3(11.52)=34.56

Since we have rectangle ABCD , AD=BC .

Thus, AF+FD=AD=BC

BC=11.52+34.56=46.08

Now, we can use cosine to find the length of .

$\sin38=\frac{9}{AE}$

$AE=\frac{9}{\sin38}=14.62$

Next, use the Pythagorean Theorem to find the length of .

$ED=\sqrt{(EF)^2+(FD)^2}$

$ED=\sqrt{9^2+34.56^2}=35.71$

Now that we have all the side lengths of the pentagon, we can find its perimeter.

$\text{Perimeter}=AB+BC+CD+DE+EA$

$\text{Perimeter}=25+46.08+25+35.71+14.62=146.41$

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Example Question #7 : How To Find The Perimeter Of A Pentagon

In pentagon ABCDE below, the length of is one-third the length of . Find the perimeter of the pentagon.

Possible Answers:

271.58

273.26

280.69

263.41

Correct answer:

273.26

Explanation:

Notice that this pentagon is made up of two right triangles and one rectangle.

Start by finding the length of . Since we are given an angle measurement and a length of a leg of the triangle, we can use tangent to find the length of .

$\tan 26=\frac{13}{AF}$

$AF=\frac{13}{\tan 26}=26.65$

Now, we know that is three times the length of .

FD=3(AF)=3(26.65)=79.95

Since we have rectangle ABCD , AD=BC .

Thus, AF+FD=AD=BC

BC=26.65+79.95=106.60

Now, we can use sine to find the length of .

$\sin 26=\frac{13}{AE}$

$AE=\frac{13}{\sin 26}=29.66$

Next, use the Pythagorean Theorem to find the length of .

$ED=\sqrt{(EF)^2+(FD)^2}$

$ED=\sqrt{13^2+79.95^2}=81.00$

Now that we have all the side lengths of the pentagon, we can find its perimeter.

$\text{Perimeter}=AB+BC+CD+DE+EA$

$\text{Perimeter}=28+106.60+28+81.00+29.66=273.26$

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

In the pentagon below, the length of is one-third that of . Find the length of diagonal .

Possible Answers:

80.50

81.72

85.99

79.38

Correct answer:

80.50

Explanation:

Start by using the Pythagorean theorem to find the length of .

$AF=\sqrt{(AE)^2-(EF)^2}$

$AF=\sqrt{22^2-18^2}=\sqrt{324}$

Since is one-third the length of , we know that the length of must be $3\sqrt{340}$ .

Now, notice that the pentagon is made up of one rectangle and two right triangles.

Focus on rectangle ABCD . We can find the length of by adding together the lengths of and .

$\text{AD}=\sqrt{324}+3\sqrt{324}=4\sqrt{324}$

Since we are working with a rectangle, we know that AD=BC . Thus, we can use the Pythagorean Theorem again to find the length of .

$AC=\sqrt{(AB)^2+(BC)^2}$

$AC=\sqrt{36^2+(4\sqrt{324})^2}=\sqrt{6480}=80.50$

Make sure to round to two places after the decimal.

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

What is the length of a diagonal of a pentagon with side length of 20 centimeters (to the nearest tenth)?

Possible Answers:

$32.4\;cm$

$34.4\;cm$

$30.0\;cm$

$34.4\;cm$

Correct answer:

$32.4\;cm$

Explanation:

Let be the length fo a diagonal.

One interior angle of a pentagon measures

$C=\frac{180(5-2)}{5}=108^{\circ }$

A diagonal of this pentagon, along with two sides, forms an isosceles triangle with sides 20, 20 and ; the side of length is opposite the angle of measure $108^{\circ }$ , and each side of length 20 is opposite an angle of measure $\frac{180-108}{2}=36^{\circ }$ .

The Law of Sines helps us to determine :

$\frac{d }{sin\;108^{\circ}}=\frac{20 }{sin\;36^{\circ}}$

$d=\frac{20\cdot sin\;108^{\circ} }{sin\;36^{\circ}}\approx\frac{20\cdot 0.9511 }{0.5878}\approx 32.4$

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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 $\small 125$ . Find the length of one side of the pentagon.

Possible Answers:

$\small 25$

$\small 20$

$\small 35$

$\small 75$

$\small 15$

Correct answer:

$\small 25$

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.

$\small 125=5(s)$

$\small s=\frac{125}{5}=25$

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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 $\small \small 75$ . Find the length of one side of the pentagon.

Possible Answers:

$\small 20$

$\small 12$

$\small 25$

$\small 5$

$\small 15$

Correct answer:

$\small 15$

Explanation:

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Intermediate Geometry : Pentagons

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

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

Example Question #3 : How To Find The Perimeter Of A Pentagon

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

Example Question #5 : How To Find The Perimeter Of A Pentagon

Example Question #6 : How To Find The Perimeter Of A Pentagon

Example Question #7 : How To Find The Perimeter Of A Pentagon

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

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

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

All Intermediate Geometry Resources

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