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## SAT (Fall 2023)

### Course: SAT (Fall 2023) > Unit 6

Lesson 6: Additional Topics in Math: lessons by skill- Volume word problems | Lesson
- Right triangle word problems | Lesson
- Congruence and similarity | Lesson
- Right triangle trigonometry | Lesson
- Angles, arc lengths, and trig functions | Lesson
- Circle theorems | Lesson
- Circle equations | Lesson
- Complex numbers | Lesson

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# Circle theorems | Lesson

## What are circle theorems problems, and how frequently do they appear on the test?

Circle theorems problems are all about finding ,
, and angles in circles.

In this lesson, we'll learn to:

- Use central angles to calculate arc lengths and sector areas
- Calculate angle measures in circles

On your official SAT, you'll likely see

**1 question**on circle theorems.Part of this lesson builds upon the Congruence and similarity skill.

**Note:**All angle measures in this lesson are in degrees. To learn more about

**radians**, check out the Angles, arc lengths, and trig functions lesson.

**You can learn anything. Let's do this!**

## How do I use central angles to calculate arc lengths and sector areas?

### Arc length from central angle

### Area of a sector

### The relationship between central angle, arc length, and sector area

**Good news:**You

*do not*need to remember the formulas for the circumference and area of a circle for the SAT! At the beginning of each SAT math section, the following relevant information is provided as reference.

Description | Formula/quantity |
---|---|

Circumference of a circle | C, equals, 2, pi, r |

Area of a circle | A, equals, pi, r, squared |

Number of degrees of arc in a circle | 360 |

A

**central angle**in a circle is formed by two radii. This angle lets us define a portion of the circle's circumference (an**arc**) or a portion of the circle's area (a**sector**).The number of degrees of arc in a circle is 360. Since the circumference and the area both describe the full 360, degrees arc of the circle, we can set up proportional relationships between parts and wholes of any circle to solve for missing values:

#### Let's look at some examples!

In the figure above, O is the center of the circle. If the area of the circle is 16, pi, what is the area of the shaded region?

In the figure above, point A is the center and the length of arc B, C, start superscript, \frown, end superscript is start fraction, 3, divided by, 10, end fraction of the circumference of the circle. What is the value of x ?

### Try it!

## How do I find angle measures in circles?

### Angle relationships in circles

Sometimes we'll be asked to apply our knowledge of angle relationships to angles within a circle. In additional to common angle relations theorems, the questions will also ask us to use two important circle-related facts.

The first we've already covered in the previous section: the sum of central angle measures in a circle is 360, degrees.

The second is that since all radii have the same length, any triangle that contains two radii is an

**isosceles triangle**.For example, in the figure above, start overline, O, A, end overline and start overline, O, C, end overline are radii of the circle, so O, A, equals, O, C. Triangle A, O, C is an isosceles triangle, and the measures of angle, O, A, C and angle, O, C, A are both 30, degrees.

#### Let's look at an example!

In the figure above, O is the center of the circle. What is the value of x ?

### Try it!

## Your turn!

## Things to remember

## Want to join the conversation?

- is it necessary to memorize sector area formula?(16 votes)
- not necessarily. If you intuitively understand the relationships (that the central angle is a fraction of 360 degrees, that the arc length is a fraction of the circumference, and that the sector area is a fraction of the whole area), then the formula doesn't have to be practicularly memorized.(18 votes)

- Under "Try it" there is no answer of "congruent"! Please advise!(6 votes)
- There is no answer of "congruent", which the angles in the question are, but there is an answer that they're "vertical angles". Vertical angles originate from the same point, and use the same two straight lines for their rays in such a way that they face exactly opposite from each other. You can prove that a pair of vertical angles has the same measure because they are both supplementary to the same angle. When taking the actual SAT, even though you can think of a true statement that fits the math problem, if its not in the answer choices then it doesn't really matter. You have to get your answer from the answer choices, and the people that make the SAT practically never make mistakes.(12 votes)

- In the circles therom problem, there are equations such as sector area=1/2*sector radians * radius^2. Why are these equations not shown in this lesson.(2 votes)
- Who knows? Maybe it's an oversight, or maybe the author wanted you to make the connection yourself. The article implicitly states the formula in the section titled "The relationship between central angle, arc length, and sector area" when it says how the area of a sector is the fraction of that area of the circle, and the area of a circle is pi*r^2.

Area of a circle = pi*r^2

Fraction of circle that is the sector = theta / 2pi

If you multiply these equations together, you get that the area of the sector is theta/2 * r^2, just like you said.(6 votes)