1 MATH 3A CHAPTER 10 POWERPOINT PRESENTATION CIRCLES AND SPHERES
2 LEARNING TARGETS AFTER YOU COMPLETE THIS CHAPTER, YOU WILL BE ABLE TO: IDENTIFY FORMULAS FOR: CIRCUMFERENCE, DIAMETER, RADIUSSOLVE PROBABILITY PROBLEMSDETERMINE THE AREA OF A CIRCLEDEFINE TRIGONOMETRIC RATIOS AND USE THEMDETERMINE VOLUME AND SURFACE AREA OF A SPHERE
3 CIRCLE - VOCABULARYCIRCLE: SET OF POINTS AT THE SAME DISTANCE FROM A GIVEN POINTRADIUS: (r) DISTANCE BETWEEN THE CENTER OF A CIRCLE AND ANY POINT ON THE CIRCLECHORD: LINE SEGMENT JOINING TWO POINTS ON A CIRCLEDIAMETER: (d) A CHORD THAT PASSES THROUGH THE CENTER OF A CIRCLECIRCUMFERENCE: (c)THE COMPLETE LENGTH AROUND A CIRCLEQUADRANT: ONE-FOURTH OF A CIRCLE
12 Area and ProbabilityProbability means the chances or likelihood of an event happening.Suppose you pick any point inside the circle, what is the probability of picking a point in the top semi-circle? 1 out of 2If a circle is split into four quadrants, what is the probability of landing on a quadrant with an even number: 1 out of 2
13 Area = πr² (use when you know the radius) Area of a CircleArea = πr² (use when you know the radius)Area = ¼πd² (use when you know the diameter)
14 ADDITIONAL CIRCLE VOCABULARY SECTOR: THE AREA ENCLOSED WITHIN A CENTRAL ANGLE OF A CIRCLE CENTRAL ANGLE: AN ANGLE WITH ITS VERTEX AT THE CENTER OF A CIRCLE AND THE CIRCLE’S RADII (PLURAL OF RADIUS) AS ITS SIDE ARC: A PORTION OF A CIRLCE BOUNDED BY TWO DISTINCT POINTS ON THE CIRCLE
19 More Circle Vocabulary Inscribed Angle: An angle formed by two chords that intersect on the circle.Intercepted Arc: The arc of a circle within an inscribed angle.Tangent: A line that touches but does not intersect a circle.Point of Tangency: The point where the tangent touches the circle
22 CIRLCE VOCABULARY, CONTINUES… Perpendicular Bisector: A set of points equidistant from two given points.Equidistant: At an equal distance.Locus of Points: A set of points that satisfy a certain condition.
26 And the Circle Vocabulary Just Keeps Coming!!!! Circumcircle: A circle that passes through three vertices of a triangle.Circumcenter: Center of a circumcircle and located at the intersection of the perpendicular bisectors of any two sides of a triangle.Angle Bisector: Locus of points equidistant from the sides of an angle.Incircle: A circle inside a triangle and tangent to each of the triangle’s sides.
27 What That Looks Like!Circumcircle & Circumcenter:
30 Sine, Cosine, Tangent Unit Circle: Circle whose radius is one. Sine (sin): for an angle of a right triangle, not the right angle, the ratio of the length of the opposite leg divided by the length of the hypotenuse.Cosine (cos): for an angle of a right triangle, not the right angle, the ratio of the length of the adjacent leg divided by the length of the hypotenuse.Tangent (tan): for an angle of a right triangle, not the right angle, the ratio of the length of the opposite side divided by the length of the adjacent leg.
31 TrigonometryTrigonometry – The branch of mathematics dealing with the relation between the sides and angles of triangles. For right triangles.
32 The SphereSphere: Locus of points in space equidistant from a fixed point.Great Circle: Circle on a sphere whose center is the center of the sphere and whose radius equals the radius of the sphere.Hemisphere: Half of a sphere.Poles: Endpoints of the diameter of a sphere.Formulas: SA = 4πr² Volume: 4/3πr³
33 What the Parts of a Sphere Look Like Great Circle: