Presentation on theme: "Chapter 12: Area of Shapes"— Presentation transcript:
1Chapter 12: Area of Shapes 12.1: Area of Rectangles
2Area of Rectangles The Area of an L-unit by W-unit rectangle is Area = L x WTrue for any non-negative values L and W
3Section 12.2: Moving and Additive Principles About Area
4The 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 rotationsAdditive 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.
5Example ProblemsEx 1: Determine the area of the following shape.
12Example ProblemEx 1: Determine the area ofthe following triangle.
13Triangle DefinitionsDef: The base of a triangle is any of its three sidesDef: Once the base is selected, the height is the line segment thatis perpendicular to the base &connects the base or an extension of it to the opposite vertex
20Definitions for Parallelograms Def: The base of a parallelogram is any of its four sidesDef: Once the base is selected, the height of a parallelogram is a line segment thatperpendicular to the base &connects the base or an extension of it to a vertex on not on the base
21Area of a Parallelogram The area of a parallelogram with base b and height h is𝐴𝑟𝑒𝑎 = 𝑏∙ℎ
23What is shearing? Def: The process of shearing a polygon: Pick a side as its baseSlice the polygon into infinitesimally thin strips that are parallel to the baseSlide strips so that they all remain parallel to and stay the same distance from the base
25Result of ShearingCavalieri’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 baseThe strips remain the same lengthThe height of the stacked strips remains the same
27DefinitionsDef: The circumference of a circle is the distance around a circleRecall: radius- the distance from the center to any point onthe circlediameter- the distance across the circle throughthe center𝐷=2𝑟
28The number πDef: The number pi, or π, is the ratio of the circumference and diameter of any circle. That is,𝜋= 𝐶 𝐷Circumference Formulas: The circumference of a circle is given by𝐶=π∙D or 𝐶=2πr
29Quick Example ProblemEx 1: A circular racetrack with a radius of 4 miles has what length for each lap?
30How to demonstrate the size of π See activities 12M and 12N
31Area of a CircleThe area of a circle with radius 𝑟 is given by𝐴=π∙ 𝑟 2See Activity 12O to see why
32Example ProblemsEx 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?
33Section 12.7: Approximating Areas of Irregular Shapes
34How do we estimate the area of the following shape?
35Methods for Estimating Area Graph Paper:Draw/trace shape onto graph paperCount the approximate number of squares inside the shapeConvert the number of squares into a standard unit of area based on the size of each squareModeling Dough:Cover the shape with a layer (of uniform thickness) of modeling doughReform the dough into a regular shape such as a rectangle or circle (of the same thickness)Calculate the area of the regular shapeCard Stock:Draw/trace shape onto card stockCut out the shape and measure its weightWeigh a single sheet of card stockUse ratios of the weights and the area of one sheet to estimate the shape’s area