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Course: Digital SAT Math > Unit 13
Lesson 4: Circle theorems: advancedCircle 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 . Since the circumference and the area both describe the full 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, is the center of the circle. If the area of the circle is , what is the area of the shaded region?
In the figure above, point is the center and the length of arc is of the circumference of the circle. What is the value of ?
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 .
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, and are radii of the circle, so . Triangle is an isosceles triangle, and the measures of and are both .
Let's look at an example!
In the figure above, is the center of the circle. What is the value of ?
Try it!
Your turn!
Things to remember
Want to join the conversation?
- Circles are so pointless(217 votes)
- that's a good one xD(6 votes)
- im taking the SAT on the 11th. Wish me luck(129 votes)
- Me too! Break your leg bro! God bless :)(83 votes)
- It's gonna be alright. We will all do well.(81 votes)
- i thought this was going to be hard and complex but this was a breeze to learn(43 votes)
- annd therin lies the end. whew. stares in disbelief
gl y'all :')(23 votes) - what are the things we need to take on test day?my exam is on aug26(10 votes)
- how was the test? You mind sharing score?(1 vote)
- Since when SAT became this difficult?(14 votes)
- rule of three is gonna save us all(13 votes)
- can i bring my own calculator for the digital sat?(9 votes)
- Hey you, yes you reading this. Even if it seems hard just keep trying and you'll do great. You got this!(11 votes)