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, RADIUS
SOLVE PROBABILITY PROBLEMS
DETERMINE THE AREA OF A CIRCLE
DEFINE TRIGONOMETRIC RATIOS AND USE THEM
DETERMINE VOLUME AND SURFACE AREA OF A SPHERE

3. 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

4. WHAT THAT LOOKS LIKE

5. CHORDS

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

7. Circle Formulas
Area:

8. Circumference
Circumference of a circle:

9. 2 formulas (if you know the radius or the diameter)

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

11. Symbol for approximately:

12. 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

13. Area = πr² (use when you know the radius)
Area of a Circle
Area = 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

15. WHAT DOES THAT LOOK LIKE?

16. Sector/Segment/Quadrant

17. Central Angles

18. ARCS

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

20. Tangent & Point of Tangency

21. Inscribed Angle & Intercepted Arc

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.

23. Perpendicular Bisector

24. Equidistant Points

25. Locus of Points

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:

28. Angle Bisectors

29. Incircle

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

32. 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

33. What the Parts of a Sphere Look Like
Great Circle:

34. Hemispheres

35. Poles