GMAT Math : Calculating the percentage of a sector from an angle

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Example Questions

Example Question #1 : Geometry

A sector of a circle has a central angle equal to 45 degrees. What percentage of the circle is comprised by the sector?

Possible Answers:

$10\%$ $\displaystyle 10\%$

$12.5\%$ $\displaystyle 12.5\%$

$6\%$ $\displaystyle 6\%$

$25\%$ $\displaystyle 25\%$

Correct answer:

$12.5\%$ $\displaystyle 12.5\%$

Explanation:

The entire circle is 360 degrees, therefore we can set up proportions and cross multiply.

$\frac{x}{100}=\frac{45}{360}$ $\displaystyle \frac{x}{100}=\frac{45}{360}$

$\displaystyle 360x=4500$

x=12.5 $\displaystyle x=12.5$

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Example Question #1 : Geometry

Consider the Circle $\displaystyle O$ :

Circle3

(Figure not drawn to scale.)

Suppose sector BOX $\displaystyle BOX$ covers an area of $\small 27 \pi\:m^2$ $\displaystyle \small 27 \pi\:m^2$ . What percentage of the area of the circle does sector BOX $\displaystyle BOX$ cover?

Possible Answers:

$0.12\%$ $\displaystyle 0.12\%$

$120\%$ $\displaystyle 120\%$

$27\%$ $\displaystyle 27\%$

$12\%$ $\displaystyle 12\%$

$1.2\%$ $\displaystyle 1.2\%$

Correct answer:

$12\%$ $\displaystyle 12\%$

Explanation:

To find the percentage of the area of the circle that sector BOX $\displaystyle BOX$ covers, divide the area of sector BOX $\displaystyle BOX$ by the total area of the circle:

Area of the circle:

$\small A=\pi r^2=\pi (15\:m)^2=225 \pi\:m^2$ $\displaystyle \small A=\pi r^2=\pi (15\:m)^2=225 \pi\:m^2$

Percentage:

$\frac{27 \pi \:m^2}{225 \pi \:m^2}=\frac{27}{225}=\frac{3}{25} =0.12$ $\displaystyle \frac{27 \pi \:m^2}{225 \pi \:m^2}=\frac{27}{225}=\frac{3}{25} =0.12$

To go from a decimal to a percent, multiply by 100 $\displaystyle 100$ . This gets us to $12\%$ $\displaystyle 12\%$ , the correct answer.

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Example Question #1 : Geometry

Circle T represents a round birthday cake. If the first slice will have a central angle of $\displaystyle 80$ degrees, what percentage of the total cake is in the first slice?

Possible Answers:

$22.2\%$ $\displaystyle 22.2\%$

$29.2\%$ $\displaystyle 29.2\%$

$11.1\%$ $\displaystyle 11.1\%$

$80.8\%$ $\displaystyle 80.8\%$

$44.4\%$ $\displaystyle 44.4\%$

Correct answer:

$22.2\%$ $\displaystyle 22.2\%$

Explanation:

In this question, the slice of cake can be thought of as a sector. We are given that its central angle is 80 degrees and asked to find what percentage of the whole it represents. Straightforward division is all we need here. We are not give a radius or any way of finding one. All we need to find is the percentage of the whole. To do that, recall that a circle has 360 degrees total and compute the following:

$\frac{80}{360}=\frac{8}{36}=\frac{2}{9}=.2\bar{2}$ $\displaystyle \frac{80}{360}=\frac{8}{36}=\frac{2}{9}=.2\bar{2}$

From here multiply by 100 to get the percentage.

$0.22\cdot 100=22.2\%$ $\displaystyle 0.22\cdot 100=22.2\%$

So the first slice represents about 22.2% of the total cake!

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Example Question #2 : Geometry

If a sector has an angle of $18^{\circ}$ $\displaystyle 18^{\circ}$ , what percentage of the circle's area is covered by the sector?

Possible Answers:

$18\%$ $\displaystyle 18\%$

$10\%$ $\displaystyle 10\%$

$5\%$ $\displaystyle 5\%$

$9\%$ $\displaystyle 9\%$

$36\%$ $\displaystyle 36\%$

Correct answer:

$5\%$ $\displaystyle 5\%$

Explanation:

The percentage of a circle covered by a sector is equal to the angle of the sector divided by the full measure of the circle, $360^{\circ}$ $\displaystyle 360^{\circ}$ . The given sector has an angle of $18^{\circ}$ $\displaystyle 18^{\circ}$ , so whatever percent this is of $360^{\circ}$ $\displaystyle 360^{\circ}$ will tell us what percent of the circle's area is covered by the sector:

$\frac{18^{\circ}}{360^{\circ}}=\frac{1}{20}=0.05=5\%$ $\displaystyle \frac{18^{\circ}}{360^{\circ}}=\frac{1}{20}=0.05=5\%$

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Example Question #241 : Gmat Quantitative Reasoning

What percentage of a circle is a sector if the angle of the sector is $27^{\circ}$ $\displaystyle 27^{\circ}$ ?

Possible Answers:

$10\%$ $\displaystyle 10\%$

$7.5\%$ $\displaystyle 7.5\%$

$27\%$ $\displaystyle 27\%$

$5\%$ $\displaystyle 5\%$

$13.5\%$ $\displaystyle 13.5\%$

Correct answer:

$7.5\%$ $\displaystyle 7.5\%$

Explanation:

The full measure of a circle is $360^{\circ}$ $\displaystyle 360^{\circ}$ , so any sector will cover whatever fraction of the circle that its angle is of $360^{\circ}$ $\displaystyle 360^{\circ}$ . We are given a sector with an angle of $27^{\circ}$ $\displaystyle 27^{\circ}$ , so this sector will cover a percentage of the circle equal to whatever fraction $27^{\circ}$ $\displaystyle 27^{\circ}$ is of $360^{\circ}$ $\displaystyle 360^{\circ}$ . This gives us:

$\frac{27^{\circ}}{360^{\circ}}=\frac{3}{40}=0.075=7.5\%$ $\displaystyle \frac{27^{\circ}}{360^{\circ}}=\frac{3}{40}=0.075=7.5\%$

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Example Question #2 : Calculating The Percentage Of A Sector From An Angle

What percentage of a circle's total area is covered by a sector with an angle of $54^{\circ}$ $\displaystyle 54^{\circ}$ ?

Possible Answers:

$27\%$ $\displaystyle 27\%$

$12.5\%$ $\displaystyle 12.5\%$

$54\%$ $\displaystyle 54\%$

$10\%$ $\displaystyle 10\%$

$15\%$ $\displaystyle 15\%$

Correct answer:

$15\%$ $\displaystyle 15\%$

Explanation:

The full measure of a circle is $360^{\circ}$ $\displaystyle 360^{\circ}$ , so any sector will cover whatever fraction of the circle that its angle is of $360^{\circ}$ $\displaystyle 360^{\circ}$ . We are given a sector with an angle of $54^{\circ}$ $\displaystyle 54^{\circ}$ , so this sector will cover a percentage of the circle equal to whatever fraction $54^{\circ}$ $\displaystyle 54^{\circ}$ is of $360^{\circ}$ $\displaystyle 360^{\circ}$ . This gives us:

$\frac{54^{\circ}}{360^{\circ}}=\frac{3}{20}=0.15=15\%$ $\displaystyle \frac{54^{\circ}}{360^{\circ}}=\frac{3}{20}=0.15=15\%$

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GMAT Math : Calculating the percentage of a sector from an angle

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #1 : Geometry

Example Question #1 : Geometry

Example Question #1 : Geometry

Example Question #2 : Geometry

Example Question #241 : Gmat Quantitative Reasoning

Example Question #2 : Calculating The Percentage Of A Sector From An Angle

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