Circles - GMAT Quantitative

Card 0 of 712

Question

What is the area of a $60^{\circ}$ sector of a circle?

Statement 1: The diameter of the circle is 48 inches.

Statement 2: The length of the arc is $8 \pi$ inches.

Answer

The area of a $60^{\circ}$ sector of radius is

$A= \frac{60}{360} \pi r^{2} = \frac{1}{6} \pi r^{2}$

From the first statement alone, you can halve the diameter to get radius 24 inches.

From the second alone, note that the length of the $60^{\circ}$ arc is

$L= \frac{60}{360} \cdot 2 \pi r = \frac{\pi r}{3}$

Given that length, you can find the radius:

$8 \pi = \frac{\pi r}{3}$

Either way, you can get the radius, so you can calculate the area.

The answer is that either statement alone is sufficient to answer the question.

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Question

The above figure shows two quarter circles inscribed inside a rectangle. What is the total area of the white region?

Statement 1: The area of the black region is $50\pi$ square centimeters.

Statement 2: The rectangle has perimeter 60 centimeters.

Answer

The width of the rectangle is equal to the radius of the quarter circles, which we call ; the length is twice that, or .

The area of the rectangle is $r \cdot 2r = 2r^{2}$ ; the total area of the two black quarter circles is $2 \cdot \frac{1}{4} \cdot \pi r^{2} = \frac{1}{2} \pi r^{2}$ , so the area of the white region is their difference,

$2r^{2} - \frac{1}{2} \pi r^{2}$

Therefore, all that is needed to find the area of the white region is the radius of the quarter circle.

If we know that the area of the black region is $50 \pi$ centimeters, then we can deduce using this equation:

$\frac{1}{2} \pi r^{2} = 50 \pi$

If we know that the perimeter of the rectangle is 60 centimeters, we can deduce via the perimeter formula:

Either statement alone allows us to find the radius and, consequently, the area of the white region.

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Question

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

Statement 1: The sector with central angle $\angle BOD$ has area $48 \pi$ .

Statement 2: $\widehat{BD }= 8 \pi$ .

Answer

Assume Statement 1 alone. No clues are given about the measure of $\angle BOD$ , so that of $\angle AOB$ , and, subsequently, the area of the shaded sector, cannot be determined.

Assume Statement 2 alone. Since the circumference of the circle is not given, it cannot be determined what part of the circle $\widehat{BD }$ , or, subsequently, $\widehat{AB}$ , is, and therefore, the central angle of the sector cannot be determined. Also, no information about the area of the circle can be determined.

Now assume both statements are true. Let be the radius of the circle and $N ^{\circ }$ be the measure of $\angle BOD$ . Then:

$\frac{N}{360} \cdot \pi r ^{2} = 48 \pi$

and

$\frac{N}{360} \cdot 2 \pi r = 8 \pi$

The statements can be simplified as

$r ^{2}N = 17,280$

and

From these two statements:

$\frac{r ^{2}N }{rN}=\frac{ 17,280}{1,440}$

; the second statement can be solved for :

.

$m \angle BOD = 120^{\circ }$ , so $m \angle AOB = 60^{\circ }$ .

Since , the circle has area $A = \pi r^{2} = \pi \cdot 12^{2} = 144 \pi$ . Since we know the central angle of the shaded sector as well as the area of the circle, we can calculate the area of the sector as

$\frac{1}{6} \times 144 \pi = 24\pi$ .

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Question

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

Statement 1: $m \angle BCD = 120^{\circ }$ .

Statement 2: The circle has circumference $28 \pi$ .

Answer

To find the area of a sector of a circle, we need a way to find the area of the circle and a way to find the central angle $\angle BOD$ of the sector.

Statement 1 alone gives us the circumference; this can be divided by $2 \pi$ to yield radius $28 \pi \div 2 \pi = 14$ , and that can be substituted for in the formula $A = \pi r^{2}$ to find the area: $A = \pi \cdot 14^{2} = 196 \pi$ .

However, it provides no clue that might yield $m \angle BOD$ .

From Statement 2 alone, we can find $m \angle BOD$ . $\angle BCD$ , an inscribed angle, intercepts an arc twice its measure - this arc is $\widehat{BAD}$ , which has measure $240^{\circ }$ . $\widehat{B D}$ , the corresponding minor arc, will have measure $360 ^{\circ}- m \widehat{BAD} = 360 ^{\circ} - 240 ^{\circ} = 120 ^{\circ}$ . This gives us $m \angle BOD$ , but no clue that yields the area.

Now assume both statements are true. The area is $196 \pi$ and the shaded sector is $\frac{120^{\circ }}{360^{\circ }}= \frac{1}{3}$ of the circle, so the area can be calculated to be $\frac{1}{3} \times 196 \pi = \frac{196 \pi}{3}$ .

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Question

The circle in the above diagram has center . Give the ratio of the area of the white sector to that of the shaded sector.

Statement 1:

Statement 2: $m \angle ACB = 30 ^{\circ }$

Answer

We are asking for the ratio of the areas of the sectors, not the actual areas. The answer is the same regardless of the actual area of the circle, so information about linear measurements such as radius, diameter, and circumference is useless. Statement 2 alone is unhelpful.

Statement 1 alone asserts that $m \angle ACB = 30 ^{\circ }$ . $\angle ACB$ is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle $\angle AOB$ that intercepts it - has twice its measure, or $2 \times 30^{\circ }= 60 ^{\circ }$ . From angle addition, this can be subtracted from $180^{\circ }$ to yield the measure of central angle $\angle BOD$ of the shaded sector, which is $120^{\circ }$ . That makes that sector $\frac{120^{\circ }}{360^{\circ }} = \frac{1}{3}$ of the circle. The white sector is $\frac{2}{3}$ of the circle, and the ratio of the areas can be determined to be $\frac{2}{3} : \frac{1}{3}$ , or .

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Question

The circle in the above diagram has center . Give the ratio of the area of the white sector to that of the shaded sector.

Statement 1: $m \angle ACB = 30 ^{\circ }$

Statement 2: $m \angle BOD = 120 ^{\circ }$

Answer

Statement 1 alone asserts that $m \angle ACB = 30 ^{\circ }$ . This is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle $\angle AOB$ that intercepts it - has twice this measure, or $60 ^{\circ }$ .

Statement 2 alone asserts that $m \angle BOD = 120 ^{\circ }$ . By angle addition, $m\angle AOB = 180 ^{\circ } - m \angle BOD = 180 ^{\circ } - 120 ^{\circ } = 60 ^{\circ }$ .

Either statement alone tells us that the shaded sector is $\frac{60^{\circ }}{360^{\circ }}= \frac{1}{6}$ of the circle, and that the white sector is $\frac{5}{6}$ of it; it can be subsequently calculated that the ratio of the areas is $\frac{5}{6} : \frac{1}{6}$ , or .

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Question

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

Statement 1: The circle has circumference $40 \pi$ .

Statement 2: $m \angle ACB = 30 ^{\circ }$

Answer

To find the area of a sector of a circle, we need a way to find the area of the circle and a way to find the central angle of the sector.

Statement 1 alone gives us the circumference; this can be divided by $2 \pi$ to yield the radius, and that can be substituted for in the formula $A = \pi r^{2}$ to find the area. However, it provides no clue that might yield $m \angle AOB$ .

Statement 2 alone asserts that $m \angle ACB = 30 ^{\circ }$ . This is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle that intercepts it - has twice this measure, or $60 ^{\circ }$ . Therefore, Statement 2 alone gives the central angle, but does not yield any clues about the area.

Assume both statements are true. The radius is $40 \pi \div 2 \pi = 20$ and the area is $A = \pi \cdot 20^{2 }= 400 \pi$ . The shaded sector is $\frac{60^{\circ }}{360^{\circ }}= \frac{1}{6}$ of the circle, so the area can be calculated to be $\frac{1}{6} \times 400 \pi = \frac{200 \pi}{3}$ .

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Question

What is the circumference of a circle?

(1)The diameter of this circle is 10

(2)The area of this circle is $25\pi$

Answer

The calculation of the circumference is $C=2\pi r$ . From statement (1) we know that $d=10$ . Therefore $r=5$ , and we can calculate $C$ using the formula. From statement (2) we know that $\pi r^{2}=25\pi$ . Therefore $r=5$ , and we can calculate $C$ using the formula.

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Question

If and are points on a plane and lies inside the circle with center and radius 5, does lie inside circle ?

(1) The length of line segment is 7.

(2) The length of line segment is 7.

Answer

(1) The max distance between two points in the circle is twice the length of the

raidus (diameter = $2 \times 5$ = ). However, can still be anywhere on the plane

(outside of the circle) as the statement does not indicate otherwise. Therefore, this statement is insuffieicent.

(2) The length of the line segment from to is greater than the radius of the circle. Thus, must be outside of the circle. This statement is sufficient.

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Question

Find the diameter of circle B.

I) Circle B has a circumference of $\small 8\pi m$ .

II) Circle B has an area of $\small 16\pi m^2$ .

Answer

We are given the area and circumference of a circle and asked to find the diameter.

Given the following equations:

$\small a=\pi r^2$

$\small c=2\pi r$

$\small r=2d$

We can see that having either area or circumference will allow us to find the radius and in turn the diameter.

Thus, either statement is sufficient by itself.

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Question

Find the diameter of circle $\small \lambda$

I) The area of $\frac{1}{6}$ of $\small \lambda$ is $54\pi$ .

II) An arc making up $\frac{1}{6}^{th}$ of $\small \lambda$ is $6\pi$ .

Answer

I) Gives us a portion of the area of the circle. From this we can find the total area and solve for the radius. Then we can double our answer to find the diameter.

II) Gives us a portion of the circumference. From this we can find the total circumference and work our way back to the radius and then the diameter.

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Question

The circle of center is inscribed in square . What is the diameter of the circle?

(1) The ratio of the diameter to the circumference of the circle is $\frac{1}{\pi}$ .

(2) The area of square is .

Answer

To find the diameter of the circle we would need information about the square or about the circle itself.

Statement 1 gives us a ratio of the diameter to the circumference $d\pi$ of the circle.

If we write the equation it is $\frac{d}{d\pi}$ or $\frac{1}{\pi}$ .

Therefore, for all circles, the ratio of the diameter to the circumference will be $\frac{1}{\pi}$ .

This statement is not helping.

Statement 2 on the other hand gives us the area of the square and therefore allows us to calculate the side of the circle, which is the same as the diameter.

Hence, statement 2 alone is sufficient.

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Question

How far has the tip of the hour hand of a clock traveled since noon?

It is now 5:00 PM.

The hour hand is half the length of the minute hand.

Answer

The time alone is insufficient without the length of the hand. The second statement does not give us that information, only the relationship between the lengths of the two hands, which is useless without the length of the minute hand.

The answer is that both statements together are insufficient to answer the question.

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Question

Arc $\widehat{AB}$ is located on circle $\odot O$ ; $\widehat{CD}$ is located on $\odot P$ Which arc, if either, is longer?

Statement 1: $OA = 2 \cdot PC$

Statement 2: $m \angle AOB = m \angle CPD$

Answer

$\angle AOB$ and $\angle CPD$ are the central angles that intercept $\widehat{AB}$ and $\widehat{CD}$ , respectively. The measure of an arc is equal to that of its central angle, so, is we are given Statement 2, that $m \angle AOB = m \angle CPD$ , we know that $m \widehat{AB} = m \widehat{CD}$ . The arcs are the same portion of their respective circles. The larger of $\odot O$ and $\odot P$ determines which arc is longer; this is given in Statement 1, since, if $OA = 2 \cdot PC$ , then $\odot O$ has the greater radius and circumference.

Both statements together are sufficient to show that $\widehat{AB}$ is the longer of the two, but neither alone is suffcient. From Statement 1, the relative sizes of the circles are known, but not the degree measures of the arcs; it is possible for an arc on a larger circle to have length less than, equal to, or greater than the arc on the smaller circle. From Statement 2 alone, the degree measures of the arcs can be proved equal, but not thier lengths.

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Question

Arc $\widehat{AB}$ is located on circle $\odot O$ ; $\widehat{CD}$ is located on $\odot P$ . Which arc, if either, has greater degree measure?

Statement 1: $\widehat{AB}$ and $\widehat{CD}$ have the same length.

Statement 2:

Answer

The arc with the greater degree measure is the one which is the greater part of its circle.

If both arcs have the same length, then the one that is the greater part of its circle must be the one on the smaller circle; Statement 1 alone tells us both have the same length, but not which circle is smaller.

If , then $\odot O$ has the greater radius and, sqbsequently, the greater circumference; it is the larger circle. But we know nothing about the measures of the arcs, so Statement 2 alone is insufficient.

If we know both statements, however, we know that, since the arcs have the same length, and $\odot O$ is the larger circle - with greater circumference - $\widehat{AB}$ must be take up the lesser portion of its circle, and have the lesser degree measure of the two.

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Question

Note: Figure NOT drawn to scale.

In the above figure, is the center of the circle. Give the length of arc $\widehat{AB}$ .

Statement 1: $m \angle ABO = 35 ^{\circ }$

Statement 2: $n = 110 ^{\circ }$

Answer

If either or both Statement 1 and Statement 2 are known, then then only thing about $\widehat{AB}$ that can be determined is that it is an arc of measure $110 ^{\circ }$ . Without knowing any of the linear measures of the circle, such as the radius or the circumference, it is impossible to determine the length of $\widehat{AB}$ .

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Question

Note: Figure NOT drawn to scale.

In the above figure, is the center of the circle. Give the length of arc $\widehat{AB}$ .

Statement 1: $\Delta OAB$ is an equilateral triangle.

Statement 2: $\Delta OAB$ has area $25 \sqrt{3}$ .

Answer

From Statement 1 alone, $m \angle AOB = 60 ^{\circ }$ , so $\widehat{AB}$ can be determined to be a $60 ^{\circ }$ arc. But no method is given to find the length of the arc.

From Statement 2 alone, neither $m \angle AOB$ nor radius can be determined, as the area of a triangle alone cannot be used to determine any angle or side.

From the two statements together, $m \angle AOB = 60 ^{\circ }$ , and the common sidelength of the equilateral triangle can be determined from the formula

$A = \frac{s^{2} \sqrt{3}}{4}$

This sidelength is the radius of the circle. Once is calculated, the circumference can be calculated, and the arc length will be $\frac{60}{360} = \frac{1}{6}$ of this.

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Question

Note: Figure NOT drawn to scale.

In the above figure, is the center of the circle. Give the length of arc $\widehat{AB}$ .

Statement 1:

Statement 2: Major arc $\widehat{ACB}$ has length $40 \pi$ .

Answer

Statement 1 only establishes that $\widehat{AB}$ is one-third of the circle. Without other information such as the radius, the circumference, or the length of an arc, it is impossible to determine the length of the chord. Statement 2 alone is also insufficient to give the length of the chord, for similar reasons.

The two statements together, however, establish that $40 \pi$ is the length of the major arc of a $120^{\circ }$ central angle, and therefore, two-thirds the circumference. The circumference can therefore be calculated to be $40 \pi \div \frac{2}{3} = 60 \pi$ , and minor arc $\widehat{AB}$ is one third of this, or $20 \pi$ .

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Question

Note: Figure NOT drawn to scale.

In the above figure, is the center of the circle, and $\overline{AO} \perp \overline{BO}$ . Give the length of arc $\widehat{AB}$ .

Statement 1: $AB = 10 \sqrt{2}$ .

Statement 2: The area of $\Delta AOB$ is .

Answer

$\overline{AO} \perp \overline{BO}$ , making $\Delta AOB$ a right triangle. Since $\overline{AO }\cong \overline{BO}$ , both segments being radii, $\Delta AOB$ is also a 45-45-90 triangle.

From Statement 1 alone, it can be determined by way of the 45-45-90 Theorem that $AO = AB \div \sqrt{2} = 10 \sqrt{2} \div \sqrt{2} = 10$ .

From Statement 2 alone, since the area of a right triangle is half the product of its legs,

$\frac{1}{2} (AO)(OB) = A$

Since and ,

$\frac{1}{2} (AO) ^{2} = 50$

$(AO) ^{2} = 100$

From either statement alone, the radius of the circle can be calculated. From there, the circumference can be calculated, and the length of the arc can be found by multiplying the circumference by $\frac{90}{360} = \frac{1}{4}$

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Question

Data Sufficiency Question

Calculate the area of a circle.

1. The radius of the circle is 4.

2. The circumference of the circle is 24.

Answer

The area of a circle can be calcuated using the equation:

$A=r^2\pi$

and the circumference calculated using:

$C=2r\pi$

The radius is the only information required for calculating the area of a circle and that can be obtained from the circumference, therefore, either statement is sufficient.

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