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Digital SAT Math

Course: Digital SAT Math > Unit 5

Lesson 4: Circle theorems: foundations
Test prep
Digital SAT Math
Foundations: Geometry and trigonometry
Circle theorems: foundations
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Circle theorems | Lesson

Google Classroom
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
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

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
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).
A central angle in a circle defines the area of a sector and the length of an arc.
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!

A circle has center O. A sector with a right central angle is shaded.
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?
A circle has center A, and points B and C are on the circle. The three points are also vertices of triangle ABC, and the measure of angle BAC is x degrees.
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
A circle has center O and radii OA and OC. The measure of central angle AOC is 150 degrees. The arc defined by the central angle is bolded, and the sector defined by the central angle is shaded.
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.
A circle has center O and radii OA and OC. The two radii and chord AC form an isosceles triangle. The measure of angle AOC is 120 degrees, and the measures of angles OAC and OCA are both 30 degrees.
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!

A circle has center O and radii OA and OC. AOC is a triangle, and the measure of angle OAC is 50 degrees. The central angle of the circle, excluding the measure of angle AOC, measure x degrees.
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
A circle has center O and diameters AC and BD. Triangles ABO and CDO are inside the circle. In triangle ABO, angle ABO measures 35 degrees. In triangle CDO, the angle CDO measures x degrees.
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
A circle has center O and two radii, OA and OC. The central angle AOC measures 120 degrees.
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
A circle has center O and two radii, OA and OC. The minor sector formed by O, OA, and OC is shaded and has a central angle of x degrees.
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
A circle has center O, radius OB, and diameter AC. The radii OB, OC, and the chord BC form a triangle, and angle OCB in the triangle measures 25 degrees. Central angle AOB measures x degrees.
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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