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SAT Math : Sectors

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Example Question #1 : How To Find The Length Of An Arc

Secant

Figure NOT drawn to scale

Refer to the above figure. Evaluate .

Possible Answers:

t = 14

t = 84

t = 56

t = 28

t = 21

Correct answer:

t = 28

Explanation:

$\overline{NB}$ and $\overline{ND }$ , being secant segments of a circle, intercept two arcs such that the measure of the angle that the segments form is equal to one-half the difference of the measures of the intercepted arcs - that is,

$\frac{1}{2} (m \overarc{BD} - m \overarc {AC }) = m \angle BND$

Setting $m \overarc{BD} = 3t , m \overarc {AC } = t , m \angle BND = 28^{\circ }$ and solving for :

$\frac{1}{2} (3t - t ) = 28$

$\frac{1}{2} (2t ) = 28$

t= 28

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Example Question #41 : Geometry

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?

Possible Answers:

4.5°

45°

12.5°

90°

25°

Correct answer:

45°

Explanation:

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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Example Question #44 : Geometry

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?

Possible Answers:

300

280

240

260

180

Correct answer:

240

Explanation:

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

360/12 = 240 degrees

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Example Question #1 : How To Find The Angle Of A Sector

Circle2

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.

Possible Answers:

75^o

60^o

110^o

30^o

80^o

Correct answer:

30^o

Explanation:

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.

AD+AB+CD+BC=360^o

AD+AB+80^o+70^o=360^o

AD+AB=210^o

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

2AB+AB=210^o

3AB=210^o

AB=70^o

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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Example Question #1 : How To Find The Angle Of A Sector

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 ?

Possible Answers:

30^o

120^o

60^o

45^o

90^o

Correct answer:

90^o

Explanation:

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 90^o.

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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Example Question #1 : How To Find The Angle Of A Sector

Circle

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?

Possible Answers:

cannot be determined

100

Correct answer:

Explanation:

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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Example Question #2 : How To Find The Angle Of A Sector

What is the angle of a sector of area in^2 on a circle having a radius of $15\:in$ ?

Possible Answers:

$15.22^{\circ}$

$0.06^{\circ}$

$22.92^{\circ}$

$3.00^{\circ}$

$72.00^{\circ}$

Correct answer:

$22.92^{\circ}$

Explanation:

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 360 degrees in a circle:

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

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

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Example Question #3 : How To Find The Angle Of A Sector

What is the angle of a sector that has an arc length of 13.5 on a circle of diameter ?

Possible Answers:

$35.81^{\circ}$

$14.24^{\circ}$

$10.74^{\circ}$

$194.14^{\circ}$

$128.92^{\circ}$

Correct answer:

$128.92^{\circ}$

Explanation:

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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Example Question #31 : Circles

Inscribed angle

Figure NOT drawn to scale.

Refer to the above diagram. $\overarc {ABC}$ is a semicircle. Evaluate $m \overarc {AB}$ .

Possible Answers:

$88^{\circ }$

$66 ^{\circ }$

$55^{\circ }$

$77 ^{\circ }$

Insufficient information is given to answer the question.

Correct answer:

$66 ^{\circ }$

Explanation:

An inscribed angle of a circle that intercepts a semicircle is a right angle; therefore, $\angle B$ , which intercepts the semicircle $\overarc{ABC}$ , is such an angle. Consequently, $m \angle B = 90 ^{\circ }$ , and $\bigtriangleup ABC$ is a right triangle. The acute angles of $\bigtriangleup ABC$ are complementary, so

$m \angle A + m \angle C = 90 ^{\circ }$

(7t+1 )+ (4t+1) = 90

7t+4t+1+1 = 90

11t+2 = 90

11t+2 -2 = 90 - 2

11t = 88

$11t\div 11 = 88 \div 11$

t = 8

The measure of inscribed $\angle C$ is

$m \angle C = (4t+1 )^{\circ } = (4 \cdot 8+1 )^{\circ } = 33 ^{\circ }$ .

An inscribed angle of a circle intercepts an arc of twice its degree measure, so

$m \overarc {AB} = 2 \cdot m \angle C= 2 \cdot 33^{\circ } = 66 ^{\circ }$ .

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Example Question #1 : How To Find The Angle Of A Sector

Secant 2 Figure NOT drawn to scale

Refer to the above figure. $\overline{AB}$ is a diameter of the circle. Evaluate .

Possible Answers:

t = 34

t = 17

t = 28

t = 56

t = 68

Correct answer:

t = 56

Explanation:

$\overline{AB}$ is a diameter, so $\overarc{BC A }$ is a semicircle, and

$m \overarc{BC A } = 180^{\circ }$ ,

or, equivalently,

$m \overarc{BC} + m \overarc {AC } = 180^{\circ }$

In terms of , since $m \overarc {AC } = t^{\circ }$ ,

$m \overarc{BC} + t ^{\circ }= 180^{\circ }$

$m \overarc{BC} + t^{\circ } - t ^{\circ }= 180 ^{\circ } - t^{\circ }$

$m \overarc{BC} =( 180 - t)^{\circ }$

$\overline{NB}$ and $\overline{NC }$ , being a secant segment and a tangent segment to a circle, respectively, intercept two arcs such that the measure of the angle that the segments form is equal to one-half the difference of the measures of the intercepted arcs - that is,

$\frac{1}{2} (m \overarc{BC} - m \overarc {AC }) = m \angle CNB$

Setting $m \overarc{BC} = \left (180 - t \right )^{\circ }$ , $m \overarc {AC } = t^{\circ }$ , and $m \angle CNB = 34^{\circ }$ :

$\frac{1}{2}\left [ (180 - t) - t \right ]= 34$

$\frac{1}{2} (180 - 2t) = 34$

$\frac{1}{2} \cdot 180 -\frac{1}{2} \cdot 2t = 34$

90 - t= 34

90-t-90 = 34 -90

-t =- 56

t = 56

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SAT Math : Sectors

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #1 : How To Find The Length Of An Arc

Example Question #41 : Geometry

Example Question #44 : Geometry

Example Question #1 : How To Find The Angle Of A Sector

Example Question #1 : How To Find The Angle Of A Sector

Example Question #1 : How To Find The Angle Of A Sector

Example Question #2 : How To Find The Angle Of A Sector

Example Question #3 : How To Find The Angle Of A Sector

Example Question #31 : Circles

Example Question #1 : How To Find The Angle Of A Sector

All SAT Math Resources

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