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Chapter 12: Area of Shapes

Chapter 12: Area of Shapes. 12.1: Area of Rectangles. Area of Rectangles. The Area of an L-unit by W-unit rectangle is Area = L x W True for any non-negative values L and W. Section 12.2: Moving and Additive Principles About Area. The Moving and Additive Principles.

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Chapter 12: Area of Shapes

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  1. Chapter 12: Area of Shapes • 12.1: Area of Rectangles

  2. Area of Rectangles • The Area of an L-unit by W-unit rectangle is • Area = L x W • True for any non-negative values L and W

  3. Section 12.2: Moving and Additive Principles About Area

  4. The Moving and Additive Principles • Moving Principle: If you move a shape rigidly without stretching it, then its area does not change. • Rigid motions include translations, reflections, and rotations • Additive Principle: If you combine a finite number of shapes without overlapping them, then the area of the resulting shape is the sum of the areas of the individual shapes.

  5. Example Problems • Ex 1:Determine the area of the following shape.

  6. Ex 1:

  7. Ex 2: Determine the • area of the following • shape.

  8. Ex 3: The UK Math Department is going to retile hallway of the 7th floor of POT, shown below. How many square feet is the hallway?

  9. See Activity 12 C, problem 1

  10. Ex 4: Determine the area of the following hexagon.

  11. Section 12.3: Area of Triangles

  12. Example Problem • Ex 1: Determine the area of the following triangle.

  13. Triangle Definitions • Def: The base of a triangle is any of its three sides • Def: Once the base is selected, the height is the line segment that • is perpendicular to the base & • connects the base or an extension of it to the opposite vertex

  14. Base and Height Ex’s

  15. Area of a Triangle • The area of a triangle with base b and height h is given by the formula • It doesn’t matter which side you choose as the base!

  16. Revisiting Example 1 • Ex 1: Determine the area of the following triangle.

  17. See problems in Activities 12F and 12G

  18. Section 12.4: Area of Parallelograms and Other Polygons

  19. See Activity 12H

  20. Definitions for Parallelograms • Def: The base of a parallelogram is any of its four sides • Def: Once the base is selected, the height of a parallelogram is a line segment that • perpendicular to the base & • connects the base or an extension of it to a vertex on not on the base

  21. Area of a Parallelogram • The area of a parallelogram with base b and height h is

  22. Section 12.5: Shearing

  23. What is shearing? Def: The process of shearing a polygon: • Pick a side as its base • Slice the polygon into infinitesimally thin strips that are parallel to the base • Slide strips so that they all remain parallel to and stay the same distance from the base

  24. Examples of Shearing

  25. Result of Shearing • Cavalieri’s Principle: The original and sheared shapes have the same area. Key observations during the shearing process: • Each point moves along a line parallel to the base • The strips remain the same length • The height of the stacked strips remains the same

  26. Section 12.6: Area of Circles and the Number π

  27. Definitions • Def: The circumference of a circle is the distance around a circle • Recall: radius- the distance from the center to any point on the circle diameter- the distance across the circle through the center

  28. The number π • Def: The number pi, or π, is the ratio of the circumference and diameter ofanycircle. That is, • Circumference Formulas: The circumference of a circle is given by or

  29. Quick Example Problem • Ex 1: A circular racetrack with a radius of 4 miles has what length for each lap?

  30. How to demonstrate the size of π • See activities 12M and 12N

  31. Area of a Circle • The area of a circle with radius is given by • See Activity 12O to see why

  32. Example Problems • Ex 2: If you make a 5 foot wide path around a circular courtyard that has a 15 foot radius, what is the area of the new path? • Ex 3: A mile running track has the following shape consisting of a rectangle with 2 semicircles on the ends. If you are planting sod inside the track, how many square feet of sod do you need?

  33. Section 12.7: Approximating Areas of Irregular Shapes

  34. How do we estimate the area of the following shape?

  35. Methods for Estimating Area • Graph Paper: • Draw/trace shape onto graph paper • Count the approximate number of squares inside the shape • Convert the number of squares into a standard unit of area based on the size of each square • Modeling Dough: • Cover the shape with a layer (of uniform thickness) of modeling dough • Reform the dough into a regular shape such as a rectangle or circle (of the same thickness) • Calculate the area of the regular shape • Card Stock: • Draw/trace shape onto card stock • Cut out the shape and measure its weight • Weigh a single sheet of card stock • Use ratios of the weights and the area of one sheet to estimate the shape’s area

  36. See Example problems in Activity 12Q

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