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### Course: Digital SAT Math > Unit 13

Lesson 4: Circle theorems: advanced# 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 | |

Area of a circle |

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 ${360}^{\circ}$ 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 $\stackrel{\u2322}{BC}$ is $\frac{3}{10}$ 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 ${360}^{\circ}$ .

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, $\stackrel{\u2015}{OA}$ and $\stackrel{\u2015}{OC}$ are radii of the circle, so $OA=OC$ . Triangle $AOC$ is an isosceles triangle, and the measures of $\mathrm{\angle}OAC$ and $\mathrm{\angle}OCA$ are both ${30}^{\circ}$ .

#### 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?

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