Circles - PSAT Math

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Question

A circle with an area of 13_π_ in2 is centered at point C. What is the circumference of this circle?

Answer

The formula for the area of a circle is A = _πr_2.

We are given the area, and by substitution we know that 13_π_ = _πr_2.

We divide out the π and are left with 13 = _r_2.

We take the square root of r to find that r = √13.

We find the circumference of the circle with the formula C = 2_πr_.

We then plug in our values to find C = 2√13_π_.

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Question

A 6 by 8 rectangle is inscribed in a circle. What is the circumference of the circle?

Answer

First you must draw the diagram. The diagonal of the rectangle is also the diameter of the circle. The diagonal is the hypotenuse of a multiple of 2 of a 3,4,5 triangle, and therefore is 10.
Circumference = π * diameter = 10_π_.

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Question

Ashley has a square room in her apartment that measures 81 square feet. What is the circumference of the largest circular area rug that she can fit in the space?

Answer

In order to solve this question, first calculate the length of each side of the room.

The length of each side of the room is also equal to the length of the diameter of the largest circular rug that can fit in the room. Since $C=\pi d$ , the circumference is simply

$C=9\pi$

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Question

A gardener wants to build a fence around their garden shown below. How many feet of fencing will they need, if the length of the rectangular side is 12 and the width is 8?

Answer

The shape of the garden consists of a rectangle and two semi-circles. Since they are building a fence we need to find the perimeter. The perimeter of the length of the rectangle is 24. The perimeter or circumference of the circle can be found using the equation C=2π(r), where r= the radius of the circle. Since we have two semi-circles we can find the circumference of one whole circle with a radius of 4, which would be 8π.

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Question

If a circle has an area of $81\pi$ , what is the circumference of the circle?

Answer

The formula for the area of a circle is πr2. For this particular circle, the area is 81π, so 81π = πr2. Divide both sides by π and we are left with r2=81. Take the square root of both sides to find r=9. The formula for the circumference of the circle is 2πr = 2π(9) = 18π. The correct answer is 18π.

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Question

A car tire has a radius of 18 inches. When the tire has made 200 revolutions, how far has the car gone in feet?

Answer

If the radius is 18 inches, the diameter is 3 feet. The circumference of the tire is therefore 3π by C=d(π). After 200 revolutions, the tire and car have gone 3π x 200 = 600π feet.

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Question

A circle has the equation below. What is the circumference of the circle?

(x – 2)2 + (y + 3)2 = 9

Answer

The radius is 3. Yielding a circumference of $6\pi$ .

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Question

The diameter of a circle is defined by the two points (2,5) and (4,6). What is the circumference of this circle?

Answer

We first must calculate the distance between these two points. Recall that the distance formula is:√((x2 - x1)2 + (y2 - y1)2)

For us, it is therefore: √((4 - 2)2 + (6 - 5)2) = √((2)2 + (1)2) = √(4 + 1) = √5

If d = √5, the circumference of our circle is πd, or π√5.

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Question

If a circle (shown above) with area $\small 36\pi$ is divided into 6 equal slices, what is the arc length of one of the slices?

Note: The above figure is not necessarily drawn to scale.

Answer

Begin by solving for the circumference of the circle. Use the area of the circle, which is given, and the equation for the area of a circle to determine the radius of the circle:

$\small \pi r^2$ = $\small 36\pi$

Divide both sides by $\small \pi$ .

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

$\small r^2 = 36$

Solve for :

$\small \sqrt{r^2}=\sqrt{36}$

$\small r=6$

The radius of the circle is 6. Now find the circumference.

Circumference is equal to 2 times the radius multiplied by $\small \pi$ .

$\small C=2r\pi$

$\small C = 2(6)\pi$

$\small C = 12\pi$

Now that we have the circumference, divide by 6 to find the length of one of the slices of the circle:

$\small 12\pi/6 = 2\pi$

The arc length of one of the slices of the circle is $\small 2\pi$ .

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Question

In the circle above, the length of arc BC is 100 degrees, and the segment AC is a diameter. What is the measure of angle ADB in degrees?

Answer

Since we know that segment AC is a diameter, this means that the length of the arc ABC must be 180 degrees. This means that the length of the arc AB must be 80 degrees.

Since angle ADB is an inscribed angle, its measure is equal to half of the measure of the angle of the arc that it intercepts. This means that the measure of the angle is half of 80 degrees, or 40 degrees.

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Question

The length of an arc, , of a circle is $8\pi$ and the radius, , of the circle is . What is the measure in degrees of the central angle, $\theta$ , formed by the arc ?

Answer

The circumference of the circle is $2\pi r$ .

$2\pi(16)=32\pi$

The length of the arc S is $8\pi$ .

A ratio can be established:

$\frac{S}{\text{circumference}}=\frac{\theta}{360^o}$

$\frac{8\pi}{32\pi}=\frac{\theta}{360^o}$

Solving for _ $\theta$ _yields 90o.

Note: This makes sense. Since the arc S was one-fourth the circumference of the circle, the central angle formed by arc S should be one-fourth the total degrees of a circle.

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Question

In the figure above that includes Circle O, the measure of angle BAC is equal to 35 degrees, the measure of angle FBD is equal to 40 degrees, and the measure of arc AD is twice the measure of arc AB. Which of the following is the measure of angle CEF? The figure is not necessarily drawn to scale, and the red numbers are used to mark the angles, not represent angle measures.

Answer

The measure of angle CEF is going to be equal to half of the difference between the measures two arcs that it intercepts, namely arcs AD and CD.

$\angle CEF=\frac{1}{2}(AD-CD)$

Thus, we need to find the measure of arcs AD and CD. Let's look at the information given and determine how it can help us figure out the measures of arcs AD and CD.

Angle BAC is an inscribed angle, which means that its meausre is one-half of the measure of the arc that it incercepts, which is arc BC.

$\angle BAC=\frac{1}{2}BC$

$35^o=\frac{1}{2}BC\rightarrow BC=70^o$

Thus, since angle BAC is 35 degrees, the measure of arc BC must be 70 degrees.

We can use a similar strategy to find the measure of arc CD, which is the arc intercepted by the inscribed angle FBD.

$40^o=\frac{1}{2}CD\rightarrow CD=80^o$

Because angle FBD has a measure of 40 degrees, the measure of arc CD must be 80 degrees.

We have the measures of arcs BC and CD. But we still need the measure of arc AD. We can use the last piece of information given, along with our knowledge about the sum of the arcs of a circle, to determine the measure of arc AD.

We are told that the measure of arc AD is twice the measure of arc AB. We also know that the sum of the measures of arcs AD, AB, CD, and BC must be 360 degrees, because there are 360 degrees in a full circle.

Because AD = 2AB, we can substitute 2AB for AD.

This means the measure of arc AB is 70 degrees, and the measure of arc AD is 2(70) = 140 degrees.

Now, we have all the information we need to find the measure of angle CEF, which is equal to half the difference between the measure of arcs AD and CD.

$\angle CEF=\frac{1}{2}(AD-CD)$

$\angle CEF=\frac{1}{2}(140^o-80^o)=\frac{1}{2}(60^o)$

$\angle CEF=30^o$

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Question

A pie has a diameter of 12". A piece is cut out, having a surface area of 4.5π. What is the angle of the cut?

Answer

This is simply a matter of percentages. We first have to figure out what percentage of the surface area is represented by 4.5π. To do that, we must calculate the total surface area. If the diameter is 12, the radius is 6. Don't be tricked by this!

A = π * 6 * 6 = 36π

Now, 4.5π is 4.5π/36π percentage or 0.125 (= 12.5%)

To figure out the angle, we must take that percentage of 360°:

0.125 * 360 = 45°

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Question

Eric is riding a Ferris wheel. The Ferris wheel has 18 compartments, numbered in order clockwise. If compartment 1 is at 0 degrees and Eric enters compartment 13, what angle is he at?

Answer

12 compartments further means 240 more degrees. 240 is the answer.

360/12 = 240 degrees

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Question

Note: Figure NOT drawn to scale.

In the above circle, . Give the ratio of the area of the white sector to that of the gray sector.

Answer

A $72^{\circ}$ sector is $\frac{72^{\circ }}{360^{\circ}}= \frac{1}{5}$ of the circle. The white sector is therefore $\frac{4}{5}$ of the circle, and the ratio of their areas is

$\frac{4}{5} : \frac{1}{5}$ ,

which simplifies to

.

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Question

Note: Figure NOT drawn to scale.

Refer to the above figure. The ratio of the area of the white sector to that of the gray sector is 5 to 1. Evaluate .

Answer

The ratio of the areas is 5 to 1, so the white sector is one sixth of the circle. This means that the central angle of the white sector is one sixth of $\frac{1}{6} \cdot 360^{\circ} = 60^{\circ }$ .

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Question

Note: Figure NOT drawn to scale.

The area of the gray sector in the above circle is $6 \pi$ . The area of the white sector is $42 \pi$ . Evaluate .

Answer

The total area of the circle is the sum of the areas of the white and gray sectors, or

$6 \pi + 42 \pi = 48 \pi$

The gray sector takes up

$\frac{6 \pi}{48 \pi} = \frac{1}{8}$

of the circle, so the degree measure of the gray sector is equal to

$\frac{1}{8} \times 360 ^{\circ} = 45 ^{\circ}$

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Question

Note: Figure NOT drawn to scale.

In the above circle, the length of arc $\widehat{MN}$ is $8 \pi$ , and the length of arc $\widehat{MYN}$ is $42 \pi$ . Evaluate .

Answer

The circumference of the circle is the sum of the lengths of the arcs $\widehat{MN}$ and $\widehat{MYN}$ , which is

$8 \pi + 42 \pi = 50 \pi$

$\widehat{MN}$ is therefore

$\frac{8 \pi}{50 \pi } = \frac{4}{25}$

of the circle, and its degree measure is

$\frac{4}{25} \cdot 360 ^{\circ} = 57 \frac{3}{5}^{\circ}$

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Question

What is the angle of a sector of area on a circle having a radius of ?

Answer

To begin, you should compute the complete area of the circle:

$A=\pi r^2$

For your data, this is:

$A=15^2\pi=225\pi$

Now, to find the angle measure of a sector, you find what portion of the circle the sector is. Here, it is:

$\frac{45}{225\pi}$

Now, multiply this by the total degrees in a circle:

$\frac{45}{225\pi}*360=22.918311805212$

Rounded, this is $22.92^{\circ}$ .

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Question

What is the angle of a sector that has an arc length of on a circle of diameter ?

Answer

The first thing to do for this problem is to compute the total circumference of the circle. Notice that you were given the diameter. The proper equation is therefore:

$C=\pi d$

For your data, this means,

$C=12\pi$

Now, to compute the angle, note that you have a percentage of the total circumference, based upon your arc length:

$\frac{13.5}{12\pi}*360=128.9155039044336$

Rounded to the nearest hundredth, this is $128.92^{\circ}$ .

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