Polygons, Cones, Right Circular Cylinders, and Right Rectangular Prisms

1. Polygons, Cones, Right Circular Cylinders, and Right Rectangular Prisms UMI: July 13, 2016

2. Polygons . • Simple: does not intersect itself • Closed: Curve can be drawn starting and stopping at the same point. • Segment: Straight line Polygons are simple closed curves with sides that are only segments.

3. Polygons and Non-polygons .

4. Perimeter of Polygons The perimeter of Polygons is the sum of lengths of its sides. Practice 1: What is the perimeter of the rectangle?

5. What is the perimeter of This polygon? Practice 2: Solution :

6. Graph the polygon ABCDEF, which has vertices A(-3,3) B(1,3) C(1, 1) D(4, 1) E(4, -3) F(-3, -3); What is the perimeter of the polygon, ABCDEF? Practice 3: Solution :

7. Area of Polygons The area of a polygon is the number of square units inside that polygon. Example: Find the area of the following polygons

8. The Area of a Rectangle Put the shape on a grid and count the number of squares The rectangle has an area of 15. If each square was 1 cm on a side, then the area would be 15 cm2 (15 square cm) The formula is: Area = w × hw = width, h = height The width is 5, and the height is 3, so we know w = 5 and h = 3: Area = 5 × 3 = 15

9. The Area of a Triangle The formula is: b = base, h = height

10. The Area of Popular Polygons Trapezoid (US)h = vertical height Parallelogramb = base, h = vertical height Squarea = length of side

11. Solution : Example: Sam cuts grass at $0.10 per square meter. How much does Sam earn cutting this area: Let's break the area into two parts: Part A is a square: Area of A = a2 = 20m × 20m = 400m2 Part B is a triangle. Viewed sideways it has a base of 20m and a height of 14m.Area of B = ½b × h = ½ × 20m × 14m = 140m2 Area = Area of A + Area of B = 400m2 + 140m2 = 540m2 Sam earns = $0.10 × 540m2 = $54

12. What is the area of the trapezoid? Practice 1: Solution :

13. What is the area of the octagon? Practice 2: Solution :

14. Cones A Cone is a solid that has a circular base and a single vertex. Right Cone: If the vertex is over the center of the base, it is called a right cone. Oblique Cone: is one where the vertex is not over the center of the circular base.

15. There are three dimensions of a cone. • The vertical height (or altitude) which is the perpendicular distance from the top down to the base. Use to denote it. • The radius of the circular base. Use to denote it. • The slant height which is the distance from the top, down the side, to a point on the base circumference. Use to denote it.

16. Surface Area of Cones A Cone is a solid that has a circular base and a single vertex.

17. Surface Area of Cones - Derivation A cone can be broken down into a circular base and the top sloping part.The Surface area is the sum of these two areas. Base: The base is a simple circle, So its area is given by; The Top: If we were to cut the cone up one side along the red line and roll it out flat, it would look something like the shaded pie-shaped section below

18. Surface Area of Cones - Derivation • This shaded section is actually part of a larger circle that has a radius of s, the slant height of the cone. We are able to find the area of sector. Which is a proportional the area of the circle. Circumference of Circumference of r

19. Volume of Cones The volume of a cone is given by the formula • There two ways to derivative the formula • Experiment • Use of Calculus

20. The diagram shows a cone of height 24 cm and base diameter 14 cm.  What is its volume? Practice 1: Solution :

21. The diagram shows a cone of height 5 cm and base radius 12 cm. What is the total • surface area of the cone? Practice 2: Solution :

22. Right Circular Cylinders A circular cylinder is a closed solid that has two parallel circular bases connected by a curved surface. Right Circular Cylinders: When the two circular bases are exactly over each other and the axis is a right angles to the base, this is a called a 'right circular cylinder'.

23. Surface Area of the Right Circular Cylinders

24. Derivation • The surface area of a cylinder can be found by breaking it down into three parts: • The two circles that make up the ends of the cylinder. • The side of the cylinder, which when "unrolled" is a rectangle

25. Derivation

26. Volume of the Right Circular Cylinders

27. The diagram shows a cylinder of diameter 20 units and height 9 units. Find the surface area of the cylinder. Practice 1: Solution :

28. The diagram shows a cylinder of height 4 units and diameter 4 units. What is its volume? Practice 2: Solution :

29. Right Rectangular Prisms A prism is a solid that has two faces that are parallel  and congruent. These are called the bases of the prism. • Right Rectangular Prisms • The bases are rectangles • The bases are exactly one above the other. This means that lines joining corresponding points on each base are perpendicular to the bases

30. Surface Area of the Right Rectangular Prism Which can be shortened to where: = width of a base rectangle = length of a base rectangleh = height of the prism

31. Volume of the Right Rectangular Prism

32. Find the surface area and the volume of this right rectangular prism Practice 1: Solution :

33. Find the surface area and the volume of this right rectangular prism Practice 2: Solution :

34. Thank You References: A problem solving approach to mathematics for elementary school teachers by Billstein, Libeskind, Lott Math is Fun : https://www.mathsisfun.com/ Math Open Reference: http://www.mathopenref.com/