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ISEE Upper Level Math : Pentagons

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Example Question #1 : Pentagons

A convex pentagon has four angles that measure $150^{\circ }$ each. What is the measure of the fifth angle?

Possible Answers:

$90^{\circ }$

$60^{\circ }$

$30^{\circ }$

$120^{\circ }$

This pentagon cannot exist.

Correct answer:

This pentagon cannot exist.

Explanation:

The angles of a pentagon measure a total of $180 (5-2) = 540 ^{\circ }$ . If we let the unknown angle measure be $x^{\circ }$ , then from this information:

150 + 150 + 150 + 150 + x = 540

600 + x = 540

600 + x -600 = 540-600

x = -60

Since an angle measure cannot be negative, this pentagon cannot exist.

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Example Question #2 : How To Find An Angle In A Pentagon

The measures of the angles of a pentagon are $x^{\circ }, x^{\circ }, x^{\circ }, y^{\circ }, y^{\circ }$ . If y = 100 , what is ?

Possible Answers:

$x = 90^{\circ}$

This pentagon cannot exist.

$x = 170^{\circ}$

$x = 113 \frac{1}{3}^{\circ}$

$x = 53 \frac{1}{3}^{\circ}$

Correct answer:

$x = 113 \frac{1}{3}^{\circ}$

Explanation:

The angles of a pentagon measure a total of $180 (5-2) = 540 ^{\circ }$ . From the information, we know that:

x + x + x + y + y = 540

3x + 2y = 540

If y = 100 , then

$3x + 2 \cdot 100 = 540$

3x + 200 = 540

3x + 200 - 200 = 540 - 200

3x = 340

$x = 340 \div 3$

$x = 113 \frac{1}{3}^{\circ}$

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Example Question #3 : How To Find An Angle In A Pentagon

The measures of the angles of a pentagon are $x ^{\circ }, x ^{\circ }, x ^{\circ }, y ^{\circ }, y ^{\circ }$ . If $x = 100 ^{\circ }$ , what is ?

Possible Answers:

$y = 50^{\circ}$

$y = 80^{\circ}$

$y = 30^{\circ}$

This pentagon cannot exist.

$y = 120^{\circ}$

Correct answer:

$y = 120^{\circ}$

Explanation:

The angles of a pentagon measure a total of $180 (5-2) = 540 ^{\circ }$ . From the information, we know:

x + x + x + y + y = 540

3x + 2y = 540

If $x = 100 ^{\circ }$ , then the above becomes:

$3 \cdot 100 + 2y = 540$

300 + 2y = 540

300 + 2y - 300 = 540 - 300

2y = 240

$2y \div 2 = 240 \div 2$

$y = 120^{\circ}$

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Example Question #4 : How To Find An Angle In A Pentagon

Solve for :

Question_11

Possible Answers:

x=90

x=110

x=70

x=87

Correct answer:

x=110

Explanation:

The sum of the interior angles of a pentagon can be determined by the following equation, where is the number of sides:

sum=180(n-2)=180(5-2)=180(3)=540

Therefore:

120+87+90+133+x=540

430+x=540

430+x-430=540-430

x=110

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

Find the perimeter of a pentagon with a side of length 12in.

Possible Answers:

$72\text{in}$

$60\text{in}$

$58\text{in}$

$96\text{in}$

$48\text{in}$

Correct answer:

$60\text{in}$

Explanation:

To find the perimeter of a pentagon, we will use the following formula:

$P = 5 \cdot a$

where a is the length of one side of the pentagon.

Now, we know the length of one side of the pentagon is 12in.

Knowing this, we can substitute into the formula. We get

$P = 5 \cdot 12\text{in}$

$P = 60\text{in}$

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

Find the perimeter of a pentagon with a side having a length of 21in.

Possible Answers:

$147\text{in}$

$105\text{in}$

$126\text{in}$

$84\text{in}$

$168\text{in}$

Correct answer:

$105\text{in}$

Explanation:

A pentagon has 5 sides. To find the perimeter, we will use the following formula:

P = 5a

where a is the length of a side of the pentagon.

Now, we know the length of a side of the pentagon is 21in.

Knowing this, we will substitute into the formula. We get

$P = 5 \cdot 21\text{in}$

$P = 105\text{in}$

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

Find the perimeter of a pentagon with a side having a length of 15cm.

Possible Answers:

$105\text{cm}$

$90\text{cm}$

$60\text{cm}$

$75\text{cm}$

Correct answer:

$75\text{cm}$

Explanation:

To find the perimeter of a pentagon, we will use the following formula:

$P = 5 \cdot a$

where a is the length of one side of the pentagon. Because a pentagon has 5 equal sides, we can use any side in the formula.

So, we know the length of one side of the pentagon is 15cm. So, we can substitute. We get

$P = 5 \cdot 15\text{cm}$

$P = 75\text{cm}$

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Example Question #4 : How To Find The Perimeter Of A Pentagon

A pentagon has a side of length 11in. Find the perimeter.

Possible Answers:

$88\text{in}$

$44\text{in}$

$66\text{in}$

$55\text{in}$

$77\text{in}$

Correct answer:

$55\text{in}$

Explanation:

To find the perimeter of a pentagon, we will use the following formula:

$P = 5 \cdot a$

where a is the length of one side of the pentagon.

Now, we know the length of one side of the pentagon is 11in.

Knowing this, we can substitute into the formula. We get

$P = 5 \cdot 11\text{in}$

$P = 55\text{in}$

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

Find the perimeter of a pentagon with a side having a length of 9in.

Possible Answers:

$45\text{in}$

$54\text{in}$

$63\text{in}$

$36\text{in}$

$72\text{in}$

Correct answer:

$45\text{in}$

Explanation:

A pentagon has 5 equal sides. To find the perimeter of a pentagon, we will use the following formula:

P = 5a

where a is the length of one side of the pentagon.

Now, we know a side of the pentagon has a length of 9in. So, we get

$P = 5 \cdot 9\text{in}$

$P = 45\text{in}$

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

In a pentagon in which all the angles are equal and the perimeter is 35 inches, how many inches would be the sum of 2 of the sides?

Possible Answers:

Correct answer:

Explanation:

In a pentagon in which all angles are equal, all sides will also be equal; therefore, the length of one side can be found by dividing the perimeter by 5 (the number of sides in a pentagon). Given that $\frac{35}{5}=7$ , the sum of 2 of the sides will be 7+7=14.

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ISEE Upper Level Math : Pentagons

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : Pentagons

Example Question #2 : How To Find An Angle In A Pentagon

Example Question #3 : How To Find An Angle In A Pentagon

Example Question #4 : How To Find An Angle In A Pentagon

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

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

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

Example Question #4 : How To Find The Perimeter Of A Pentagon

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

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

All ISEE Upper Level Math Resources

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