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# 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:
1. Use central angles to calculate arc lengths and sector areas
2. Calculate angle measures in circles
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

Arc length from subtended angleSee video transcript

### Area of a sector

Area of a sectorSee video transcript

### 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.
DescriptionFormula/quantity
Circumference of a circle$C=2\pi r$
Area of a circle$A=\pi {r}^{2}$
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:
$\frac{\text{central angle}}{{360}^{\circ }}=\frac{\text{arc length}}{\text{circumference}}=\frac{\text{sector area}}{\text{circle area}}$

#### 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 $\stackrel{⌢}{BC}$ is $\frac{3}{10}$ of the circumference of the circle. What is the value of $x$ ?

### Try it!

try: use circle proportions
In the figure above, point $O$ is the center of the circle.
What fraction of the area of the entire circle is the area of the shaded region?
If the length of
$\stackrel{⌢}{AC}$ is $10$, what is the circumference of the circle?

## 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, $\stackrel{―}{OA}$ and $\stackrel{―}{OC}$ 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!

try: find the measure of an angle inside a circle
In the figure above, $O$ is the center of the circle, and $\stackrel{―}{AC}$ and $\stackrel{―}{BD}$ are two diameters.
The measure of $\mathrm{\angle }BAO$ is
${}^{\circ }$.
The measure of $\mathrm{\angle }AOB$ is
${}^{\circ }$.
$\mathrm{\angle }AOB$ and $\mathrm{\angle }COD$ are
angles.
What is the value of $x$ ?

Practice: find arc length given the central angle
The circle above with center $O$ has a circumference of $12\pi$. What is the length of minor arc $\stackrel{⌢}{AC}$ ?

Practice: find central angle measure given sector area
In the figure above, point $O$ is the center and the shaded area is $\frac{3}{8}$ the area of the circle. What is the value of $x$ ?

Practice: find the measure of a central angle using angle relationships
In the figure above, $O$ is the center of the circle, and $\stackrel{―}{AC}$ is a diameter of the circle. What is the value of $x$ ?

## Things to remember

$\frac{\text{central angle}}{{360}^{\circ }}=\frac{\text{arc length}}{\text{circumference}}=\frac{\text{sector area}}{\text{circle area}}$

## Want to join the conversation?

• im taking the SAT on the 11th. Wish me luck
• Me too! Break your leg bro! God bless :)
• It's gonna be alright. We will all do well.
• let's hope so
• i thought this was going to be hard and complex but this was a breeze to learn
• Hey you, yes you reading this. Even if it seems hard just keep trying and you'll do great. You got this!
• annd therin lies the end. whew. stares in disbelief

gl y'all :')
• gl man
• Since when SAT became this difficult?
• what are the things we need to take on test day?my exam is on aug26