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
HOPE YOU ENJOYED IT