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MATH 3A CHAPTER 10 POWERPOINT PRESENTATION CIRCLES AND SPHERES

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LEARNING TARGETS AFTER YOU COMPLETE THIS CHAPTER, YOU WILL BE ABLE TO: IDENTIFY FORMULAS FOR: CIRCUMFERENCE, DIAMETER, RADIUS SOLVE PROBABILITY PROBLEMS DETERMINE THE AREA OF A CIRCLE DEFINE TRIGONOMETRIC RATIOS AND USE THEM DETERMINE VOLUME AND SURFACE AREA OF A SPHERE

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CIRCLE - VOCABULARY CIRCLE: SET OF POINTS AT THE SAME DISTANCE FROM A GIVEN POINT RADIUS: (r) DISTANCE BETWEEN THE CENTER OF A CIRCLE AND ANY POINT ON THE CIRCLE CHORD: LINE SEGMENT JOINING TWO POINTS ON A CIRCLE DIAMETER: (d) A CHORD THAT PASSES THROUGH THE CENTER OF A CIRCLE CIRCUMFERENCE: (c)THE COMPLETE LENGTH AROUND A CIRCLE QUADRANT: ONE-FOURTH OF A CIRCLE

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WHAT THAT LOOKS LIKE

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CHORDS

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THE RATIO PI Pi = 22/7 or 3.14 Its symbol is:

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Circle Formulas Area:

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Circumference Circumference of a circle:

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2 formulas (if you know the radius or the diameter)

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Estimation of Area of Circle Area estimation formula: area of a circle = 3r² Where r = radius Approximating the area of a circle:

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Symbol for approximately:

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Area and Probability Probability 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 2 If a circle is split into four quadrants, what is the probability of landing on a quadrant with an even number: 1 out of 2

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Area of a Circle Area = πr² (use when you know the radius) Area = ¼πd² (use when you know the diameter)

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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

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WHAT DOES THAT LOOK LIKE?

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Sector/Segment/Quadrant

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Central Angles

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ARCS

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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

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Tangent & Point of Tangency

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Inscribed Angle & Intercepted Arc

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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.

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Perpendicular Bisector

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Equidistant Points

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Locus of Points

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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.

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What That Looks Like! Circumcircle & Circumcenter:

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Angle Bisectors

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Incircle

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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.

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Trigonometry Trigonometry – The branch of mathematics dealing with the relation between the sides and angles of triangles. For right triangles.

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The Sphere Sphere: 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

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What the Parts of a Sphere Look Like Great Circle:

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Hemispheres

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Poles

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