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SAT (Fall 2023)

Course: SAT (Fall 2023) > Unit 6

Lesson 6: Additional Topics in Math: lessons by skill
Test prep
SAT (Fall 2023)
About the SAT Math Test
Additional Topics in Math: lessons by skill
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Circle theorems | Lesson

Google Classroom

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

Khan Academy video wrapper
Arc length from subtended angleSee video transcript

Area of a sector

Khan Academy video wrapper
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?
  • Your answer should be
  • a proper fraction, like 1, slash, 2 or 6, slash, 10
If the length of
A, C, start superscript, \frown, end superscript is 10, what is the circumference of the circle?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
degrees.
The measure of angle, A, O, B is
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
degrees.
angle, A, O, B and angle, C, O, D are
angles.
What is the value of x ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Your turn!

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 ?
Choose 1 answer:

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 ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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 ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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

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  • blobby green style avatar for user Maryam Afshar
    Posted 3 years ago. Direct link to Maryam Afshar's post “is it necessary to memori...”
    is it necessary to memorize sector area formula?
    (16 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user smc20130923
    Posted 2 years ago. Direct link to smc20130923's post “Under "Try it" there is n...”
    Under "Try it" there is no answer of "congruent"! Please advise!
    (6 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user Hecretary Bird
      Posted 2 years ago. Direct link to Hecretary Bird's post “There is no answer of "co...”
      Good Answer
      There is no answer of "congruent", which the angles in the question are, but there is an answer that they're "vertical angles". Vertical angles originate from the same point, and use the same two straight lines for their rays in such a way that they face exactly opposite from each other. You can prove that a pair of vertical angles has the same measure because they are both supplementary to the same angle. When taking the actual SAT, even though you can think of a true statement that fits the math problem, if its not in the answer choices then it doesn't really matter. You have to get your answer from the answer choices, and the people that make the SAT practically never make mistakes.
      (12 votes)
  • duskpin ultimate style avatar for user supernova34
    Posted 2 years ago. Direct link to supernova34's post “In the circles therom pro...”
    In the circles therom problem, there are equations such as sector area=1/2*sector radians * radius^2. Why are these equations not shown in this lesson.
    (2 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user Hecretary Bird
      Posted 2 years ago. Direct link to Hecretary Bird's post “Who knows? Maybe it's an ...”
      Who knows? Maybe it's an oversight, or maybe the author wanted you to make the connection yourself. The article implicitly states the formula in the section titled "The relationship between central angle, arc length, and sector area" when it says how the area of a sector is the fraction of that area of the circle, and the area of a circle is pi*r^2.
      Area of a circle = pi*r^2
      Fraction of circle that is the sector = theta / 2pi
      If you multiply these equations together, you get that the area of the sector is theta/2 * r^2, just like you said.
      (6 votes)