 SAT Reasoning: Circles By: Saif Ali AlYamahi Grade: 12.02

Contents  Introduction  SAT & Circles  Radius  Diameter  Circumference  Central Angle  Sector  Tangent  Arc Length  Video  5 SAT Question with Solutions

Introduction  A Circle is the collection of points in between a given point, called the center. The distance from the center to any point on the circle is called the radius, (r), the most important measurement in a circle. If you know a circle’s radius, you can figure out all its other characteristics.

SAT & Circles  On the SAT you will calculate both the area and circumference of circles, as well as the area and arc length of part of a circle. You will also solve problems about lines tangent to circles.

Radius  Is the distance from the center of a circle to its perimeter (r).

Diameter  Is the maximum distance from one point on the circle to another point on the circle.  The diameter is twice the length of the radius. d = 2r

Circumference  Is the perimeter of the circle, C = 2πr = πd.

Central Angle  Is the angle with endpoints located on a circle’s circumference and vertex located at the circle’s center. ∠ AOC below is a central angle.  Arc is a piece of the circumference.

Example  What is the measure of Arc ABC in circle O seen below?

Sector  Is a piece of the Area.

Example  The area of a sector in a circle with radius 3 is 3π. What is the measure of the central angle?  Use 3π for the sector area and π(3) 2 = 9π for the circle area  There are 360° in a circle so  The measure of the central angle is 120˚

Tangent  Is a line that touches the circle at one point. In the figure below AB is tangent to circle O.  When a line is tangent to a circle, the line is perpendicular to the radius at the point of tangency. Therefore, in the figure below m ∠ OAB = 90°.

Example ? AB is tangent to circle O at point A. OB = 13, and AB = 12. What is the radius of the circle? Use the Pythagorean Theorem. (OA) 2 + (AB) 2 = (OB) 2 Substitute AB = 12 and OB = 13. (OA) = 132. Solve for OA.(OA) =169 ⇒ (OA) 2 =25 ⇒ OA=5. Radius is 5.

Arc Length  An arc is a part of a circle’s circumference. An arc contains two endpoints and all the points on the circle between the endpoints. By picking any two points on a circle, two arcs are created: a major arc, which is by definition the longer arc, and a minor arc, the shorter one.  The arc length formula is

Real SAT Question 1 1. What is the area of a circle whose circumference is 12π? A. 9π B. 12π C. 16π D. 25π E. 36π

Solution for Question 1 Answer: E Substitute C = 12π in the circumference formula. C = 12π = 2πr Solve for r. 6 = r Substitute r = 6 and solve for A. A=πr 2 =π(6) 2 = 36π.

Real SAT Question 2 2. What is the diameter of a circle whose area is 4 ? A. 2 ÷ √π B. 4 ÷ √π C. 4 √π D. 2π E. 4π

Solution for Question 2 Answer: B Substitute A = 4 in the area formula and solve for r. The diameter is twice the radius

Real SAT Question 3 3. What is the area of the shaded region? A. 24π B. 60π C. 90π D. 120π E. 144π

Solution for Question 3 Answer: D The circle is 360  and the shaded region is 300. Write a fraction.

Real SAT Question 4 4. What is the area of the circle O if the length of Arc PQR is 8π ? A. 24π B. 60π C. 90π D. 120π E. 144π

Solution for Question 4 Answer: E The central angle is 120°. Find the circumference.

Real SAT Question 5 5. Point O is the center of both circles seen below. What is the area of the shaded regions? A. 100π B. 50π C. 36π D. 16π E. 4π

Solution for Question 5 Answer: B Combining the shaded regions is equal to half of the larger circle. Therefore, the area of the shaded regions is

Video

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