Sectors

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GMAT Quantitative › Sectors

Questions 1 - 10

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

Explanation

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

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

A given sector covers of a circle. What is the corresponding angle of the sector?

$252^{\circ}$

$300^{\circ}$

$222^{\circ}$

$202^{\circ}$

$272^{\circ}$

Explanation

A circle comprises $360^\circ$ , so a sector comprising of the circle will have an angle that is of .

Therefore:

$\angle=0.70(360)=\frac{7}{10}(360)=252^{\circ}$

What is the arc length for a sector with a central angle of $45^{\circ}$ if the radius of the circle is ?

$\frac{3\pi }{4}$

$\frac{\pi }{4}$

$\frac{3\pi }{2}$

$\frac{\pi }{2}$

$\pi$

Explanation

Using the formula for arc length, we can plug in the given angle and radius to calculate the length of the arc that subtends the central angle of the sector. The angle, however, must be in radians, so we make sure to convert degrees accordingly by multiplying the given angle by $\frac{\pi }{180}$ :

$S=r\theta$

$S=3(45^{\circ})(\frac{\pi }{180^{\circ}})=3(\frac{\pi }{4})=\frac{3\pi}{4}$

The arc $\widehat{AB}$ of a circle measures $90^{\circ }$ . The chord of the arc, $\overline{AB}$ , has length . Give the length of the arc $\widehat{AB}$ .

$\frac{ \pi \sqrt{2}}{4} x + \frac{ \pi \sqrt{2}}{2}$

$\frac{ \pi \sqrt{2}}{2} x + \pi \sqrt{2}$

$\frac{ \sqrt{2}}{\pi}x + \frac{ \sqrt{2}}{\pi}$

$\frac{2 \sqrt{2}}{\pi}x + \frac{4 \sqrt{2}}{\pi}$

$\pi x \sqrt{2}+ 2 \pi \sqrt{2}$

Explanation

Examine the figure below, which shows the arc and chord in question.

Circle x

If we extend the figure to depict the circle as the composite of four quarter-circles, each a $90^{\circ }$ arc, we see that is also the side of an inscribed square. A diagonal of this square, which measures $\sqrt{2}$ times this sidelength, or

$(x+2) \sqrt{2} = x\sqrt{2} + 2\sqrt{2}$ ,

is a diameter of this circle. The circumference is $\pi$ times the diameter, or

$\pi ( x\sqrt{2} + 2\sqrt{2}) = x \pi \sqrt{2} + 2 \pi \sqrt{2}$ .

Since a $90^{\circ }$ arc is one fourth of a circle, the length of arc $\widehat{AB}$ is

$\frac{1}{4}( x \pi \sqrt{2} + 2 \pi \sqrt{2})$

$= \frac{ \pi }{4} x\sqrt{2} + \frac{ \pi }{2} \sqrt{2}$

$= \frac{ \pi \sqrt{2}}{4} x + \frac{ \pi \sqrt{2}}{2}$

Sector

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: Arc $\widehat{AB}$ has length $30 \pi$ .

Statement 2: Arc $\widehat{BD}$ has length $60 \pi$ .

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation

Assume Statement 1 alone. Since the circumference of the circle is not given, it cannot be determined what part of the circle $\widehat{AB}$ is, and therefore, the central angle of the sector cannot be determined. Also, no information about the circle can be determined. A similar argument can be given for Statement 2 being insufficient.

Now assume both statements are true. Then the length of semicircle $\widehat{ABD}$ is equal to $30 \pi + 60 \pi = 90 \pi$ . The circumference is twice this, or $180 \pi$ . The radius can be calculated as $r = C \div( 2 \pi )=180 \pi \div (2 \pi) = 90$ , and the area, $\pi r^{2}= \pi \cdot 90^{2} = 8,100 \pi$ . Also, $\widehat{AB}$ is $\frac{30 \pi}{180 \pi}= \frac{1}{6}$ of the circle, and the area of the sector can now be calculated as $\frac{1}{6} \times 8,100 \pi = 1,350 \pi$ .

A teacher buys a supersized pizza for his after-school club. The super-pizza has a diameter of 18 inches. If the teacher is able to perfectly cut from the center a 36 degree sector for himself, what is the area of his slice of pizza, rounded to the nearest square inch?

Explanation

First we calculate the area of the pizza. The area of a circle is defined as $\pi r^2$ . Since our diameter is 18 inches, our radius is 18/2 = 9 inches. So the total area of the pizza is $\pi 9^2 = 81\pi = 254.469$ square inches.

Since the sector of the pie he cut for himself is 36 degrees, we can set up a ratio to find how much of the pizza he cut for himself. Let x be the area of the pizza he cut for himself. Then we know, $\frac{36^\circ}{360^\circ} = \frac{x}{254.469}$

Solving for x, we get x=25.45 square inches, which rounds down to 25.

A given sector of a circle comprises of the circle. What is the corresponding angle of the sector?

$162^{\circ}$

$192^{\circ}$

$212^{\circ}$

$142^{\circ}$

$132^{\circ}$

Explanation

A circle comprises $360^\circ$ , so a sector comprising of the circle will have an angle that is of .

Therefore:

$\angle=0.45(360)=\frac{9}{20}(360)=162^{\circ}$

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

Explanation

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

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

A giant clock has a minute hand that is eight feet long. The time is now 2:40 PM. How far has the tip of the minute hand moved, in inches, between noon and now?

$512 \pi \textrm{ in}$

$128 \pi \textrm{ in}$

$448 \pi \textrm{ in}$

$112 \pi \textrm{ in}$

$256 \pi \textrm{ in}$

Explanation

Between noon and 2:40 PM, two hours and forty minutes have elapsed, or, equivalently, two and two-thirds hours. This means that the minute hand has made $2 \frac{2}{3} = \frac{8}{3}$ revolutions.

In one revolution, the tip of an eight-foot minute hand moves $C = 2 \pi r =2 \pi \cdot 8 = 16 \pi$ feet, or $16 \pi \cdot 12 = 192 \pi$ inches.

After $\frac{8}{3}$ revolutions, the tip of the minute hand has moved $192 \pi \cdot 2 \frac{2}{3} = \frac{192}{1} \cdot \frac{8}{3} \cdot \pi = 512 \pi$ inches.