Digital SAT Math
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:
- 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.
|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).
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 . 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!
A circle has center O. A sector with a right central angle is shaded.
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?
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 is the center and the length of arc is of the circumference of the circle. What is the value of ?
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 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 is , 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 .
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, 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!
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, is the center of the circle. What is the value of ?
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, is the center of the circle, and and are two diameters.
The measure of is
The measure of is
What is the value of ?
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 has a circumference of . What is the length of minor arc ?
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 is the center and the shaded area is the area of the circle. What is the value of ?
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, is the center of the circle, and is a diameter of the circle. What is the value of ?
Things to remember
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