1. Connecting Geometric and Algebraic Representations Cheryl Olsen Shippensburg University October 11, 2005

2. Objectives Geometric figures: –Understand the concepts of length, area, volume & surface area –Recognize/represent 3-dimensional figures –Understand formulas for area and volumes of 2-dimensional & 3-dimensional figures Measurement –Solve problems involving length, area, & volume of geometric objects

3. Question 1: Imagine a cone inscribed in a cylinder of the same size, so that the base of the cone is the same as the base of the cylinder and the top of the cone touches the top of the cylinder. Imagine also a sphere inscribed in a cylinder so that the sphere touches the cylinder at the north and south pole and all the way around the equator. What is the ratio of the volumes of a cone, sphere and cylinder?

4. Question 2: Tennis balls are often packed snugly three to a can. What percent of the volume of the can do the tennis balls occupy?

5. Area of Rectangle Area of rectangle = base * height

6. Area of Parallelogram Area of parallelogram = base * height

7. Area of Triangle Area of triangle = ½ * area of parallelogram Area of triangle = ½ * base * height

8. Area of Trapezoid Area of trapezoid = ½ * area of parallelogram Area of trapezoid = ½ * (b1+b2) * h base of parallelogram = b1+b2

9. Area of a Circle

10. Cut apart your pizza and rearrange the slices so that it is a figure that we know how to find the area. Area of a Circle

11. base = ½ of circumference of circle height = radius of circle Area of Circle = ½ * circumference * radius Area of Circle = ½ * ( 2 * Pi * radius ) * radius Area of Circle = Pi * radius 2 Area of a Circle

12. Surface Area & Volume of 3-dimensional shapes Cylinder Cone Sphere

13. Activities A,B,C Groups of 4 3 groups start with Activity A, 3 groups start with Activity B, & 3 groups start with Activity C After 15 minutes we’ll rotate

14. Activity A Activity A Relationship between Volume of Cylinder and Volume of Sphere Volume of sphere is 2/3 of the volume of the cylinder.

15. Activity B Activity B Relationship between Volume of Cylinder and Volume of Cone Volume of cone is 1/3 of the volume of the cylinder.

16. Activity C Activity C Surface Area of a Sphere Surface Area = 4*Pi*radius 2

17. Volume of Cylinder (in which a cone & sphere fit inside it) Volume =

18. Volume of Sphere (which fits inside previous cylinder) Volume = 2/3 * volume of cylinder = 2/3 *

19. Volume of Cone (which fits inside previous cylinder) Volume = 1/3 * volume of cylinder = 1/3 *

20. Volume Cylinder vs. Volume Cone vs. Volume Sphere Volume of Cylinder Volume of Sphere Volume of Cone Ratios of the volumes are (Cylinder : Sphere : Cone ) 6:4:2 OR 3:2:1

21. Tennis balls are often packed snugly three to a can. What percent of the volume of the can do the tennis balls occupy?

22. Case Study Video on Volume of Cylinder

23. Cylinder Surface Area –What does a net of a cylinder look like? How does this help determine the surface area of a cylinder? –Surface Area =

24. Milk Tanker A stainless steel milk tanker in the shape of a right circular cylinder is 38 feet long and 5 feet in diameter. Determine the amount of stainless steel material needed to construct the tanker. Assume that 12% of the material you start with will be wasted in the construction process.

25. Milk Tanker Circumference = Pi * diameter 5 ft circumference 38 ft Surface Area = (38)(circumference) + 2(Pi * radius 2 ) = (38)(Pi * 5) + 2(Pi * 2.5 2 ) = (190*Pi) + (50*Pi/4) = 202.5 Pi square feet

26. Milk Tanker Surface Area = 202.5 Pi square feet ~ 636.17 square feet Assume that 12% of the material you start with will be wasted in the construction process. Since 12% of the original material will be wasted we can think of 88% of the original material = 636.17 sq ft.88 * original material = 202.5 Pi ~636.17 sq ft original material = 636.17 sq ft ~ 723 sq ft.88

27. Name that Common Solid 1.Side view and front view are triangles. Top view is a circle. 2.Side view and front view are rectangles. Top view is a circle. 3.Side view and front view are triangles. Top view is a square. Cone Cylinder Square pyramid

28. Name that Common Solid 4. Side view and front view are triangles. Top view is a rectangle. 5. Side view and front view are rectangles. Top view is a rectangle. Rectangular Prism Rectangular pyramid

29. Name that Common Solid 6. Side view, front view, and top view are all congruent squares. 7. Side view, front view, and top view are all congruent, and all triangles. Triangular Pyramid Cube