Circles - PSAT Math

Card 0 of 616

Question

In the circle above, the angle A in radians is $\frac{\pi}{4}$

What is the length of arc A?

Answer

Circumference of a Circle = $2\pi r$

Arc Length

$=\frac{Angle A}{2\pi}\ast2\pi r$

$=\frac{\frac{\pi}{4}}{2\pi}\ast2\pi r$

$=\frac{\pi}{(4)2\pi}\ast2\pi r$

$=\frac{1}{8}\ast2\pi r=\frac{\pi r}{4}$

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Question

Figure not drawn to scale.

In the figure above, circle C has a radius of 18, and the measure of angle ACB is equal to 100°. What is the perimeter of the red shaded region?

Answer

The perimeter of any region is the total distance around its boundaries. The perimeter of the shaded region consists of the two straight line segments, AC and BC, as well as the arc AB. In order to find the perimeter of the whole region, we must add the lengths of AC, BC, and the arc AB.

The lengths of AC and BC are both going to be equal to the length of the radius, which is 18. Thus, the perimeter of AC and BC together is 36.

Lastly, we must find the length of arc AB and add it to 36 to get the whole perimeter of the region.

Angle ACB is a central angle, and it intercepts arc AB. The length of AB is going to equal a certain portion of the circumference. This portion will be equal to the ratio of the measure of angle ACB to the measure of the total degrees in the circle. There are 360 degrees in any circle. The ratio of the angle ACB to 360 degrees will be 100/360 = 5/18. Thus, the length of the arc AB will be 5/18 of the circumference of the circle, which equals 2_πr_, according to the formula for circumference.

length of arc AB = (5/18)(2_πr_) = (5/18)(2_π_(18)) = 10_π_.

Thus, the length of arc AB is 10_π_.

The total length of the perimeter is thus 36 + 10_π_.

The answer is 36 + 10_π_.

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Question

If the area of a circle is $36\pi$ , then what is the length of the $120^\circ$ arc shown in the diagram?

Answer

We are given the area of the circle, but we need to find the circumference in order to find the arc length. The equation for the area of a circle is $A =\pi r^{2}$

Because we know that the area is 36 $\pi$ , we can use that equation to find the radius of the circle.

$36\pi =\pi r^{2}$

Divide both sides by $\pi$

$36 = r^{2}$

Take the square root of both sides, and see that the radius is 6.

We can now find the circumference of the circle using the formula $C=2\pi r$

$C=2\pi 6$

$C=12\pi$

Now that we know the circumference, we can set up a proportion. The length of the 120 degree arc is going to be only a portion of the total circumference of the circle. By putting the degree measure over 360 and setting it equal to x over the circumference, we can find exactly how long the arc is.

$\frac{120}{360}=\frac{x}{12\pi }$

When you multiply both sides by $12\pi$ , you find the solution:

$x=4\pi$

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Question

Consider a circle centered at the origin with a circumference of $13\pi$ . What is the x value when y = 3? Round your answer to the hundreths place.

Answer

The formula for circumference of a circle is $C=2\pi r$ , so we can solve for r:

$2\pi r=13\pi$

$\frac{2\pi r}{\pi}=\frac{13\pi}{\pi}$

$2r=13$

$r=\frac{13}{2}=6.5$

We now know that the hypotenuse of the right triangle's length is 13.5. We can form a right triangle from the unit circle that fits the Pythagorean theorem as such:

$x^2+y^2=r^2$

Or, in this case:

$x^2+3^3=6.5^2$

$x^2=42.26-9$

$x^2=33.25$

$x=\sqrt{33.25}=5.77$

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Question

In the figure above, rectangle ABCD has a perimeter of 40. If the shaded region is a semicircle with an area of 18π, then what is the area of the unshaded region?

Answer

In order to find the area of the unshaded region, we will need to find the area of the rectangle and then subtract the area of the semicircle. However, to find the area of the rectangle, we will need to find both its length and its width. We can use the circle to find the length of the rectangle, because the length of the rectangle is equal to the diameter of the circle.

First, we can use the formula for the area of a circle in order to find the circle's radius. When we double the radius, we will have the diameter of the circle and, thus, the length of the rectangle. Then, once we have the rectangle's length, we can find its width because we know the rectangle's perimeter.

Area of a circle = πr2

Area of a semicircle = (1/2)πr2 = 18π

Divide both sides by π, then multiply both sides by 2.

r2 = 36

Take the square root.

r = 6.

The radius of the circle is 6, and therefore the diameter is 12. Keep in mind that the diameter of the circle is also equal to the length of the rectangle.

If we call the length of the rectangle l, and we call the width w, we can write the formula for the perimeter as 2l + 2w.

perimeter of rectangle = 2l + 2w

40 = 2(12) + 2w

Subtract 24 from both sides.

16 = 2w

w = 8.

Since the length of the rectangle is 12 and the width is 8, we can now find the area of the rectangle.

Area = l x w = 12(8) = 96.

Finally, to find the area of just the unshaded region, we must subtract the area of the circle, which is 18π, from the area of the rectangle.

area of unshaded region = 96 – 18π.

The answer is 96 – 18π.

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Question

In a large field, a circle with an area of 144_π_ square meters is drawn out. Starting at the center of the circle, a groundskeeper mows in a straight line to the circle's edge. He then turns and mows ¼ of the way around the circle before turning again and mowing another straight line back to the center. What is the length, in meters, of the path the groundskeeper mowed?

Answer

Circles have an area of πr_2, where r is the radius. If this circle has an area of 144_π, then you can solve for the radius:

πr_2 = 144_π

r 2 = 144

r =12

When the groundskeeper goes from the center of the circle to the edge, he's creating a radius, which is 12 meters.

When he travels ¼ of the way around the circle, he's traveling ¼ of the circle's circumference. A circumference is 2_πr_. For this circle, that's 24_π_ meters. One-fourth of that is 6_π_ meters.

Finally, when he goes back to the center, he's creating another radius, which is 12 meters.

In all, that's 12 meters + 6_π_ meters + 12 meters, for a total of 24 + 6_π_ meters.

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Question

A circle has an area of 36π inches. What is the radius of the circle, in inches?

Answer

We know that the formula for the area of a circle is π_r_2. Therefore, we must set 36π equal to this formula to solve for the radius of the circle.

36π = π_r_2

36 = _r_2

6 = r

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Question

Circle X is divided into 3 sections: A, B, and C. The 3 sections are equal in area. If the area of section C is 12π, what is the radius of the circle?

Circle X

Answer

Find the total area of the circle, then use the area formula to find the radius.

Area of section A = section B = section C

Area of circle X = A + B + C = 12π+ 12π + 12π = 36π

Area of circle = where r is the radius of the circle

36π = πr2

36 = r2

√36 = r

6 = r

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Question

The specifications of an official NBA basketball are that it must be 29.5 inches in circumference and weigh 22 ounces. What is the approximate radius of the basketball?

Answer

To Find your answer, we would use the formula: C=2πr. We are given that C = 29.5. Thus we can plug in to get \[29.5\]=2πr and then multiply 2π to get 29.5=(6.28)r. Lastly, we divide both sides by 6.28 to get 4.70=r. (The information given of 22 ounces is useless)

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Question

Two concentric circles have circumferences of 4π and 10π. What is the difference of the radii of the two circles?

Answer

The circumference of any circle is 2πr, where r is the radius.

Therefore:

The radius of the smaller circle with a circumference of 4π is 2 (from 2πr = 4π).

The radius of the larger circle with a circumference of 10π is 5 (from 2πr = 10π).

The difference of the two radii is 5-2 = 3.

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Question

A circle with center (8, –5) is tangent to the y-axis in the standard (x,y) coordinate plane. What is the radius of this circle?

Answer

For the circle to be tangent to the y-axis, it must have its outer edge on the axis. The center is 8 units from the edge.

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Question

A circle has an area of $49\pi \ in^{2}$ . What is the radius of the circle, in inches?

Answer

We know that the formula for the area of a circle is πr_2. Therefore, we must set 49_π equal to this formula to solve for the radius of the circle.

49_π_ = _πr_2

49 = _r_2

7 = r

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Question

A certain circle has a circumference (in units) that is two times the area (in units squared). What is the radius of the circle?

Answer

The formula for the area of a circle is

$Area=\pi r^{2}$

The formula for the circumference of a circle is

$Circumference= 2\pi r$

Because the circumference is twice the area, your new formula is

$2\pi r=2(\pi r^{2})$

You can divide both sides by $2\pi r$ , giving you the answer: .

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Question

If it is 2:00 PM, what is the measure of the angle between the minute and hour hands of the clock?

Answer

First note that a clock is a circle made of 360 degrees, and that each number represents an angle and the separation between them is 360/12 = 30. And at 2:00, the minute hand is on the 12 and the hour hand is on the 2. The correct answer is 2 * 30 = 60 degrees.

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Question

What is the measure, in degrees, of the acute angle formed by the hands of a 12-hour clock that reads exactly 3:10?

Answer

The entire clock measures 360°. As the clock is divided into 12 sections, the distance between each number is equivalent to 30° (360/12). The distance between the 2 and the 3 on the clock is 30°. One has to account, however, for the 10 minutes that have passed. 10 minutes is 1/6 of an hour so the hour hand has also moved 1/6 of the distance between the 3 and the 4, which adds 5° (1/6 of 30°). The total measure of the angle, therefore, is 35°.

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Question

What is the measure of the smaller angle formed by the hands of an analog watch if the hour hand is on the 10 and the minute hand is on the 2?

Answer

A analog clock is divided up into 12 sectors, based on the numbers 1–12. One sector represents 30 degrees (360/12 = 30). If the hour hand is directly on the 10, and the minute hand is on the 2, that means there are 4 sectors of 30 degrees between then, thus they are 120 degrees apart (30 * 4 = 120).

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Question

It is 4 o’clock. What is the measure of the angle formed between the hour hand and the minute hand?

Answer

At four o’clock the minute hand is on the 12 and the hour hand is on the 4. The angle formed is 4/12 of the total number of degrees in a circle, 360.

4/12 * 360 = 120 degrees

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Question

If a clock reads 8:15 PM, what angle do the hands make?

Answer

A clock is a circle, and a circle always contains 360 degrees. Since there are 60 minutes on a clock, each minute mark is 6 degrees.

$\frac{360^o\ \text{total}}{60\ \text{minutes total}}=6\ \text{degrees per minute}$

The minute hand on the clock will point at 15 minutes, allowing us to calculate it's position on the circle.

$(15\ min)(6)=90^o$

Since there are 12 hours on the clock, each hour mark is 30 degrees.

$\frac{360^o\ \text{total}}{12\ \text{hours total}}=30\ \text{degrees per hour}$

We can calculate where the hour hand will be at 8:00.

$(8\ hr)(30)=240^o$

However, the hour hand will actually be between the 8 and the 9, since we are looking at 8:15 rather than an absolute hour mark. 15 minutes is equal to one-fourth of an hour. Use the same equation to find the additional position of the hour hand.

$240^o+(\frac{1}{4}\ hr)(30)$

We are looking for the angle between the two hands of the clock. The will be equal to the difference between the two angle measures.

$\angle=247.5^o-90^o=157.5^o$

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Question

What is the measure of the smaller angle between the hands of a clock if the time reads $7\textup{:}15$ ?

Answer

First, we must determine where the hands are located at this time. The minute hand will be exactly on the three. The hour hand will be a little bit (15 mins) passed the seven. A circle has $360^{\circ}$ in it, so because there are numbers on a clock's face, each number is separated by

$360^{\circ}\div12^{\circ}=30^{\circ}$

The angle between the and the is equal to $4\times30^{\circ}=120^{\circ}$

Finally, we must figure out exactly how far past the our hour hand is. If the hour hand moves $30^{\circ}$ in $60\textup{ minutes}$ , that means that it must move a half of a degree every minute. If we are $15\textup{ minutes}$ past $7\textup{:}00$ , we can do $\left ( \frac{1}{2} \right )15=7.5$

Therefore, our hour hand has moved an extra $7.5^{\circ}$ due to the $15\textup{ minutes}$ that have elapsed.

Our angle is equal to

$120^{\circ}+7.5^{\circ}=127.5^{\circ}$ .

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Question

The radius of the circle above is and $\angle A=45^{\circ}$ . What is the area of the shaded section of the circle?

Answer

Area of Circle = πr2 = π42 = 16π

Total degrees in a circle = 360

Therefore 45 degree slice = 45/360 fraction of circle = 1/8

Shaded Area = 1/8 * Total Area = 1/8 * 16π = 2π

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