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## SAT

### Course: SAT>Unit 6

Lesson 6: Additional Topics in Math: lessons by skill

# 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:
1. Use central angles to calculate arc lengths and sector areas
2. 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

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 circleC, equals, 2, pi, r
Area of a circleA, equals, pi, r, squared
Number of degrees of arc in a circle360
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:
start fraction, start text, c, e, n, t, r, a, l, space, a, n, g, l, e, end text, divided by, 360, degrees, end fraction, equals, start fraction, start text, a, r, c, space, l, e, n, g, t, h, end text, divided by, start text, c, i, r, c, u, m, f, e, r, e, n, c, e, end text, end fraction, equals, start fraction, start text, s, e, c, t, o, r, space, a, r, e, a, end text, divided by, start text, c, i, r, c, l, e, space, a, r, e, a, end text, end fraction

#### 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!

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
A, C, start superscript, \frown, end superscript 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, 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!

try: find the measure of an angle inside a circle
In the figure above, O is the center of the circle, and start overline, A, C, end overline and start overline, B, D, end overline are two diameters.
The measure of angle, B, A, O is
degrees.
The measure of angle, A, O, B is
degrees.
angle, A, O, B and angle, C, O, D 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 A, C, start superscript, \frown, end superscript ?

Practice: find central angle measure given sector area
In the figure above, point O is the center and the shaded area is start fraction, 3, divided by, 8, end fraction 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 start overline, A, C, end overline is a diameter of the circle. What is the value of x ?

## Things to remember

start fraction, start text, c, e, n, t, r, a, l, space, a, n, g, l, e, end text, divided by, 360, degrees, end fraction, equals, start fraction, start text, a, r, c, space, l, e, n, g, t, h, end text, divided by, start text, c, i, r, c, u, m, f, e, r, e, n, c, e, end text, end fraction, equals, start fraction, start text, s, e, c, t, o, r, space, a, r, e, a, end text, divided by, start text, c, i, r, c, l, e, space, a, r, e, a, end text, end fraction