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Chapter 12: Area of Shapes 12.1: Area of Rectangles

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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

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Section 12.2: Moving and Additive Principles About Area

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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.

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

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Ex 1:

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Ex 2: Determine the area of the following shape.

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

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See Activity 12 C, problem 1

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Ex 4: Determine the area of the following hexagon.

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Section 12.3: Area of Triangles

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

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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

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14 Base and Height Ex’s

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15 Area of a Triangle

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

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17 See problems in Activities 12F and 12G

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Section 12.4: Area of Parallelograms and Other Polygons

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19 See Activity 12H

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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

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21 Area of a Parallelogram

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Section 12.5: Shearing

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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 23

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Examples of Shearing 24

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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 25

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Section 12.6: Area of Circles and the Number π

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Definitions 27

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The number π 28

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

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

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Area of a Circle 31

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Example Problems 32

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Section 12.7: Approximating Areas of Irregular Shapes

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

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Methods for Estimating Area Graph Paper: 1.Draw/trace shape onto graph paper 2.Count the approximate number of squares inside the shape 3.Convert the number of squares into a standard unit of area based on the size of each square Modeling Dough: 1.Cover the shape with a layer (of uniform thickness) of modeling dough 2.Reform the dough into a regular shape such as a rectangle or circle (of the same thickness) 3.Calculate the area of the regular shape Card Stock: 1.Draw/trace shape onto card stock 2.Cut out the shape and measure its weight 3.Weigh a single sheet of card stock 4.Use ratios of the weights and the area of one sheet to estimate the shape’s area 35

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See Example problems in Activity 12Q 36

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