What is the perimeter of a regular polygon with

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VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 amFollowed by: 3 members

by VJesus12 » Fri Apr 27, 2018 1:18 am

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What is the perimeter of a regular polygon with sides of length 12 cm and internal angles measuring 144Â° each?

A. 96 cm
B. 108 cm
C. 120 cm
D. 132 cm
E. 144 cm

The OA is the option C.

How can I determine the perimeter without knowing how many sides there are? Someone helps me.

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Source: Beat The GMAT — Problem Solving | Fri Apr 27, 2018 1:18 am

Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pmFollowed by: 5 members; Thanked: 6 times

by Vincen » Fri Apr 27, 2018 3:31 am

Hi vjesus 12.

Here we need to use the following formula: $$180\left(n-2\right)=n\cdot\beta$$ where n represents the number of sides and beta represents the measure of one internal angle.

Here we have that beta is 144Âº, therefore $$180\left(n-2\right)=144\cdot n\ \ \Rightarrow\ \ 180n-360=144n\ \ \Rightarrow\ \ 36n=360\ \ \Rightarrow\ \ n=10.$$ Now, we know that the polygon has 10 sides.

The perimeter of a regular polygon is given by $$P=\left(NÂº\ of\ sides\right)\cdot\left(lenght\ \right)\ \Rightarrow\ \ P=10\cdot12 \ cm\ =120 \ cm.$$ Thus, the correct answer is the option C.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Fri Apr 27, 2018 5:25 am

VJesus12 wrote:What is the perimeter of a regular polygon with sides of length 12 cm and internal angles measuring 144Â° each?

A. 96 cm
B. 108 cm
C. 120 cm
D. 132 cm
E. 144 cm

Sum of ALL sides in an n-sided polygon = (n - 2)(180) degrees

So, EACH ANGLE in a regular n-sided polygon = (n - 2)(180)/n degrees

We're told that EACH ANGLE is 144Â°, so we can write: (n - 2)(180)/n = 144
Multiply both sides by n to get: (n - 2)(180) = 144n
Expand left side: 180n - 360 = 144n
Add 360 to both sides: 180n = 144n + 360
Subtract 144n from both sides: 36n = 360
Solve: n = 10

This tells us that the polygon has 10 sides
Since each side is the SAME LENGTH in a REGULAR polygon, the perimeter = (10)(12 cm) = 120 cm

Answer: C

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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swerve: Legendary Member; Posts: 2535; Joined: Sun Oct 29, 2017 2:04 pmFollowed by: 6 members

by swerve » Fri Apr 27, 2018 9:11 am

Sum of all interior angles is given by:
180(n-2) = n is number of sides

180(n-2) = 144*n
from here
n = 10

So perimeter becomes 12*10= 120 answer

Regards!

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[email protected]: Elite Legendary Member; Posts: 10392; Joined: Sun Jun 23, 2013 6:38 pmFollowed by: 511 members; Location: Palo Alto, CA; Thanked: 2867 times; GMAT Score:800

by [email protected] » Fri Apr 27, 2018 5:33 pm

Hi VJesus12,

We're told that a regular polygon has sides of length 12 cm and internal angles measuring 144Â° each. We're asked for the PERIMETER of that polygon. This question can be solved in a variety of different ways, but there's a great 'concept shortcut' that you can use - along with the Answer choices - to avoid most of the 'math' completely:

EVERY polygon has total degrees that are some multiple of 180 (a triangle is 180, a square is 360, a pentagon is 540, etc.). By extension, all of those totals are multiples of 10. Thus, whatever the total degree measure of THIS mystery polygon is, the total MUST end in a 0. With angles of 144 each, the two obvious ways to get a total that end in a 0 would be to multiply by 5 or by 10 - and that means the number of SIDES would be a multiple of 5 or 10. There's only one Answer that is (12cm)(a multiple of 5 or 10)....

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8137; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Mon Apr 30, 2018 3:43 pm

VJesus12 wrote:What is the perimeter of a regular polygon with sides of length 12 cm and internal angles measuring 144Â° each?

A. 96 cm
B. 108 cm
C. 120 cm
D. 132 cm
E. 144 cm

We need to determine the number of sides of this regular polygon first before we can determine its perimeter. Let's use the formula for sum of interior angles to determine the number of angles (and thus the number of sides) of the polygon. Recall that if a polygon has n sides, the total number of degrees of the interior angles is given by 180(n - 2). Since we are given that each interior angle measures 144 degrees, the total number of degrees of the interior angles is also 144n. Thus we can create the equation:

144n = 180(n - 2)

144n = 180n - 360

360 = 36n

10 = n

Thus, the perimeter is 10 x 12 = 120.

Alternate Solution:

An exterior angle of this polygon is 180 - 144 = 36 degrees. Since exterior angles of a polygon always add up to 360, this polygon has 360 / 36 = 10 angles; which also means this polygon has 10 sides. Since each side measures 12 cm, the perimeter is 12 x 10 = 120 cm.

Answer: C

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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