Chapter 10A SBM Mathematics and the Arts

1. Chapter 10A SBMMathematics and the Arts �Fundamentals of Geometry�

2. What Is Euclidean Geometry? A Greek mathematician named Euclid wrote a 13-volume textbook called Elements which summarized Greek knowledge of geometry. Euclidean geometry is the familiar geometry of lines, angles, and planes.

3. Points, Lines, and Planes! Point: has zero size. No object has zero size. Line: Plane: Line Segments:

4. Dimensions Dimension: The dimension of an object can be thought as the number of independent directions you could move if you were on the object.

5. One-Dimensional A line is one-dimensional because if you walk on a line you can only walk in one direction. (Forward and backward) Has one coordinate (x).

6. Two-Dimensional A plan is a two-dimensional object that you can move in two independent directions. (North/South, East/West) Has two coordinates (x and y).

7. Three-Dimensional A three-dimensional object is 3-D. You can move north/south, east/west, up/down. A three-dimensional point is located in space. It has three coordinates (x, y, and z).

8. Angles Angle: The intersection of two lines or line segments. Vertex: The point of intersection. A circle measures 360�.

11. Angle Cont.. Perpendicular: Two lines or line segments in a plane that meet in a right angle. Parallel: Two lines or line segments in a plane that are everywhere the same distance apart.

12. Example 1 Find the angles that subtend the following: a. A semicircle b. A quarter circle c. An eighth of a circle d. A hundredth of a circle.

13. Example 1 Answers a. An angle subtending a semicircle measures � ? 360� =180� b. An angle subtending a quarter circle measures � ? 360� = 90� c. An angle subtending an eighth of a circle measures 1/8 ? 360� = 45� d. An angle subtending a hundredth of a circle measures 1/100 ? 360� = 3.6�

14. Plane Geometry Terms Radius: The distance from one point on a circle to the circle�s center. Diameter: The distance across the circle passing through the center. Polygon: A closed shape in the plane made from straight line segments. Regular Polygon: All the sides have the same length and all interior angles are equal.

15. Examples of Regular Polygons

16. Perimeter Perimeter of a plane shape is the length of its boundary. To find the perimeter of a polygon by adding the lengths of the individual edges. The perimeter of a circle is called the circumference. The circumference of a circle is related to its diameter or radius by the universal constant called ?. Circumference of circle ? ? ? diameter ? 2 ? ? ? radius

17. Example 2 A window consists of a 4-foot by 6-foot rectangle capped by a semicircle. How much trim is needed to go around the window?

18. Example 2 Answer The trim must line the 4-foot base of the window, two 6-foot sides, and the semicircular cap. The three straight edges have a total length of 4ft ? 6ft ? 6ft ? 16ft. The perimeter of the semicircular cap is half of the circumference of a full circle with a diameter of 4 feet. � ? ? ? 4ft ? � ? 3.14 ? 4ft ? 6.3ft 16ft + 6.3ft = 22.3ft

19. Area Area of a Circle area of a circle ? ? ? radius 2 ? ? ? r 2 Area of a Rectangle area of rectangle base ? height ? b ? h Area of a Triangle area of triangle ? � ? base ? height ? � ? b ? h

20. Example 3 You have built a stairway in a new house and want to cover the space beneath the stairs with plywood. The stairway is 12 feet along its base and 9 feet tall. What is the area to be covered?

21. Example 3 Answer

22. Volume and Surface Area Formulas Object Picture Volume Surface Area Rectangular V= lwh SA = ph + 2B Prism Cube V= s 3 SA= 6 s 2 Right Circular V= ?r 2 h SA= 2?r 2 + 2?rh Cylinder Sphere V= 4/3?r 3 SA= 4?r 2

23. Example 4 A water reservoir has a rectangular base that measures 30 meters by 40 meters, and vertical walls 15 meters high. At the beginning of the summer , the reservoir was filled to capacity. At the end of the summer, the water depth was 4 meters. How much water was used?

24. Example 4 Answer The reservoir is a rectangular prism. - When the reservoir was filled at the beginning of the summer, the volume of water was: 30m ?40m ? 15m = 18,000m 3 - At the end of the summer, the amount of water remaining was: 30m ?40m ? 4m = 4800m 3 18,000m 3 - 4800m 3 = 13,200m 3

25. Scaling Laws Lengths always scale with the scale factor. Areas always scale with the square of the scale factor. Volumes always scale with the cube of the scale factor.

26. Example 5 Suppose that , magically, your size suddenly doubled; that is, your height, width, and depth doubled. For example, if you were 5 feet tall before, you now are 10 feet tall. A. By what factor has your waist size increased? B. How much more material will be required for your clothes? C. By what factor has your weight changed?

27. Example 5 Answer A. Waist size is like a perimeter. Therefore, your waist size simply doubles. B. Clothing covers surface area and therefore scales with the square of the scale factor. The scale factor by which you have grown is 2, so your surface area grows by a factor of 22 = 4. If your shirt used 2 square yards of material before, it now uses 4 ? 2 = 8 square yards of material. C. Your weight depends on your volume, which scales with the cube of the scale factor. Your new volume and new weight are therefore 23 = 8 times their old values.

28. Surface to Volume Ratio

29. Surface to volume Ratio Larger objects have smaller surface area to volume ratios than similarly proportioned small objects. Smaller objects have larger surface area to volume ratios than similarly proportioned larger objects.

30. Homework: 10A: 4, 6, 10 � 18, 24, 25, 29, 30 You must answer the questions completely and read the entire question.