SAT Math : Circles

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #1 : Circles

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?

Possible Answers:

36 + 36π

18 + 10π

36 + 20π

36 + 10π

18 + 36π

Correct answer:

36 + 10π

Explanation:

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 2πr, according to the formula for circumference.

length of arc AB = (5/18)(2πr) = (5/18)(2π(18)) = 10π.

Thus, the length of arc AB is 10π.

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

The answer is 36 + 10π.

Example Question #1 : Plane Geometry

Circle

In the circle above, the angle A in radians is 

What is the length of arc A?

Possible Answers:

Correct answer:

Explanation:

Circumference of a Circle = 

Arc Length

Example Question #2 : How To Find The Length Of An Arc

Slide1

In the figure above,  and  are diameters of the cirlce, which has a radius of . What is the sum of the lengths of arcs  and ?

Possible Answers:

Correct answer:

Explanation:

The formula for arclength is .

You know that  so  and  must both equal .

Since 

,

the sum the lengths of arcs  and  must equal .

Example Question #3 : How To Find The Length Of An Arc

Secant

Figure NOT drawn to scale

Refer to the above figure. Evaluate .

Possible Answers:

Correct answer:

Explanation:

 and , 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,

Setting  and solving for :

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

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:

12.5°

90°

25°

45°

4.5°

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°

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

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:

280

180

240

260

300

Correct answer:

240

Explanation:

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

360/12 = 240 degrees

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:

Correct answer:

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.

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.

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.

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.

 

 

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

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

Possible Answers:

Correct answer:

Explanation:

The circumference of the circle is .

The length of the arc S is .

A ratio can be established:

Solving for 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.

Example Question #51 : Circles

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:

100

40

90

cannot be determined

80

Correct answer:

40

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.

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

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

Possible Answers:

Correct answer:

Explanation:

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

For your data, this is:

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

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

Rounded, this is .

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