Problem Solving

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

Hi VJesus12,

We're told that a regular polygon has sides of length 12 cm and internal angles measuring 144° 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

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