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## Digital SAT Math

### Course: Digital SAT Math > Unit 5

Lesson 4: Circle theorems: foundations# Circle theorems | Lesson

A guide to circle theorems on the digital SAT

## What are circle theorems problems?

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

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

**radians**, check out the

**Unit circle trigonometry**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 |

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?

- im taking the SAT on the 11th. Wish me luck(59 votes)
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