GMAT Math : Polygons

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Example Question #31 : Polygons

What is the median of the measures of the angles of a nonagon (a nine-sided polygon)?

Possible Answers:

$135^{\circ }$

$120^{\circ }$

$108^{\circ }$

$140^{\circ }$

The question cannot be answered without knowing the measures of the individual angles.

Correct answer:

The question cannot be answered without knowing the measures of the individual angles.

Explanation:

The sum of the measures of the nine angles of any nonagon is calculated as follows:

$180 \cdot \left ( 9-2\right ) =180 \cdot 7= 1,260^{\circ }$

The median of an odd quantity of numbers is the number that falls in the center position when they are arranged in ascending order; for nine numbers, it will be the fifth-highest number. We now need to show that we need to know the actual numbers in order to find the median.

Case 1: Each angle measures $140 ^{\circ }$ .

The set is $\left \{ 140,140,140,140,140,140,140,140,140\right \}$ and the median is 140.

Case 2: Eight of the angles measure $139 ^{\circ }$ and one of them measures $148 ^{\circ }$ .

The set is $\left \{}139, 139, 139, 139, 139, 139, 139, 139, 148 \right \}$ and the median is 139.

In both cases, the sum of the angle measures is 1,260, but the medians differ between the two.

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Example Question #701 : Gmat Quantitative Reasoning

Pentagon

Note: Figure NOT drawn to scale.

Given Regular Pentagon PENTA . What is $m \angle APN$ ?

Possible Answers:

$80^{\circ }$

$72^{\circ }$

$76^{\circ }$

$64^{\circ }$

$60^{\circ }$

Correct answer:

$72^{\circ }$

Explanation:

Quadrilateral PNTA is a trapezoid, so $m \angle APN + m \angle A = 180 ^{\circ }$ .

$m \angle A = \frac{180 (5-2 )}{5} ^{\circ } = 108 ^{\circ }$ , so

$m \angle APN +108^{\circ } = 180 ^{\circ }$

$m \angle APN +108^{\circ } - 108^{\circ } = 180 ^{\circ } - 108^{\circ }$

$m \angle APN = 72^{\circ }$

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Example Question #461 : Geometry

The angles of a pentagon measure $x^{\circ }, \left (x + 10 \right )^{\circ }, \left (x + 10 \right )^{\circ }, \left (x + 15 \right )^{\circ }, \left (x + 25 \right )^{\circ }$ .

Evaluate .

Possible Answers:

x = 96

This pentagon cannot exist

x = 60

x = 120

x = 132

Correct answer:

x = 96

Explanation:

The sum of the degree measures of the angles of a (five-sided) pentagon is $180 (5 -2 ) ^{\circ } = 540^{\circ }$ , so we can set up and solve the equation:

$x+ \left(x + 10 \right )+ \left (x + 10 \right )+ \left(x + 15 \right )+ \left(x + 25 \right ) = 540$

5x+ 60= 540

5x+ 60- 60= 540- 60

5x = 480

$5x \div 5 = 480 \div 5$

x = 96

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Example Question #702 : Gmat Quantitative Reasoning

The measures of the angles of a pentagon are: $x ^{\circ },\left ( x+23 \right ) {^\circ},\left ( x+23 \right ) {^\circ},\left ( x+33 \right ) {^\circ},\left ( x+41 \right ) {^\circ}$

What is equal to?

Possible Answers:

x = 117

x= 120

x= 84

x = 61

x = 107

Correct answer:

x= 84

Explanation:

The degree measures of the interior angles of a pentagon total $180 \left ( 5-2\right ) ^{\circ } = 540^{\circ }$ , so

$x + \left ( x+23 \right ) +\left ( x+23 \right ) +\left ( x+33 \right ) +\left ( x+41 \right ) = 540$

5x + 120= 540

5x + 120- 120= 540- 120

5x = 420

$5x \div 5= 420 \div 5$

x= 84

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Example Question #703 : Gmat Quantitative Reasoning

What is the measure of an angle in a regular octagon?

Possible Answers:

$135$^{\circ}$$

$80$^{\circ}$$

$90$^{\circ}$$

$115$^{\circ}$$

$150$^{\circ}$$

Correct answer:

$135$^{\circ}$$

Explanation:

On octagon has sides. The word regular means that all of the angles are equal. Therefore, we can use the general equation for finding the angle measurement of a regular polygon:

$\frac{(n-2)180\,\textup{degrees}}{n}$ , where is the number of sides of the polygon.

$\frac{(8-2)180\,\textup{degrees}}{8}=135\,\textup{degrees}$ .

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Example Question #11 : Calculating An Angle In A Polygon

What is the measure of one exterior angle of a regular twenty-four sided polygon?

Possible Answers:

$25^{\circ }$

$15^{\circ }$

$18^{\circ }$

$20^{\circ }$

$24^{\circ }$

Correct answer:

$15^{\circ }$

Explanation:

The sum of the measures of the exterior angles of any polygon, one at each vertex, is $360^{\circ }$ . Since a regular polygon with twenty-four sides has twenty-four congruent angles, and therefore, congruent exterior angles, just divide:

$360 \div 24 = 15$

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Example Question #34 : Polygons

A pentagon with a perimeter of one mile has three congruent sides; one of the other sides is 100 feet longer than any of those three congruent sides, and the remaining side is 100 feet longer than that fourth side. What is the length of that longest side?

Possible Answers:

$916 \textrm{ ft}$

$1,096 \textrm{ ft}$

$1,196 \textrm{ ft}$

$1,116 \textrm{ ft}$

$996 \textrm{ ft}$

Correct answer:

$1,196 \textrm{ ft}$

Explanation:

If each of the five congruent sides has measure , then the other two sides have measures x + 100 and $\left (x + 100 \right ) +100 = x + 200$ . Add the sides to get the perimeter, which is equal to 5,280 feet, the solve for :

x + x + x + (x+100) + (x+ 200) = 5,280

5x + 300 = 5,280

5x + 300 - 300 = 5,280 - 300

5x = 4,980

$5x \div 5 = 4,980 \div 5$

x = 996

Each of the shortest sides is 996 feet long; the longest side is 996 + 200 = 1,196 feet long.

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Example Question #1 : Calculating The Length Of A Side Of A Polygon

The perimeter of a regular hexagon is one-half of a mile. Give the sidelength in inches.

Possible Answers:

$10,560 \textrm{ in}$

$5,280 \textrm{ in}$

$2,640 \textrm{ in}$

$7,920 \textrm{ in}$

$1,760 \textrm{ in}$

Correct answer:

$5,280 \textrm{ in}$

Explanation:

One mile is 5,280 feet. The perimeter of the hexagon is one-half of this, or 2,640 feet. Since each side of a regular hexagon is congruent, the length of one side is one-sixth of this, or $2,640 \div 6 = 440$ feet.

Multiply by 12 to convert to inches: $440 \times 12 = 5,280$ inches.

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Example Question #711 : Problem Solving Questions

The perimeter of a regular pentagon is one-fifth of a mile. Give its sidelength in feet.

Possible Answers:

$352\textrm{ ft}$

$176\textrm{ ft}$

$1,056\textrm{ ft}$

$422 \frac{2}{5} \textrm{ ft}$

$211 \frac{1}{5} \textrm{ ft}$

Correct answer:

$211 \frac{1}{5} \textrm{ ft}$

Explanation:

One mile is 5,280 feet. The perimeter of the pentagon is one-fifth of this, or $5,280 \div 5 = 1,056$ feet. Since each side of a regular pentagon is congruent, the length of one side is one fifth-of this, or $1,056 \div 5 = 211 \frac{1}{5}$ feet.

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Example Question #1 : Calculating The Length Of A Side Of A Polygon

The perimeter of a regular octagon is two kilometers. Give its sidelength in meters.

Possible Answers:

$250 \textrm{ m}$

$200\textrm{ m}$

$333 \frac{1}{3} \textrm{ m}$

$2,000\textrm{ m}$

$2,500 \textrm{ m}$

Correct answer:

$250 \textrm{ m}$

Explanation:

One kilometer is equal to 1,000 meters, so two kilometers comprise 2,000 meters. A regular octagon has eight sides of equal length, so divide by 8 to get the sidelength: $2,000 \div 8 = 250$ meters.

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GMAT Math : Polygons

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #31 : Polygons

Example Question #701 : Gmat Quantitative Reasoning

Example Question #461 : Geometry

Example Question #702 : Gmat Quantitative Reasoning

Example Question #703 : Gmat Quantitative Reasoning

Example Question #11 : Calculating An Angle In A Polygon

Example Question #34 : Polygons

Example Question #1 : Calculating The Length Of A Side Of A Polygon

Example Question #711 : Problem Solving Questions

Example Question #1 : Calculating The Length Of A Side Of A Polygon

All GMAT Math Resources

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