# Chapter 12: Area of Shapes

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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

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.

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

Ex 1:

Ex 2: Determine the area of the following shape.

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?

See Activity 12 C, problem 1

Ex 4: Determine the area of the following hexagon.

Section 12.3: Area of Triangles

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

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

Base and Height Ex’s

Area of a Triangle 𝐴𝑟𝑒𝑎= 1 2 ∙𝑏∙ℎ
The area of a triangle with base b and height h is given by the formula 𝐴𝑟𝑒𝑎= 1 2 ∙𝑏∙ℎ It doesn’t matter which side you choose as the base!

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

See problems in Activities 12F and 12G

Section 12.4: Area of Parallelograms and Other Polygons

See Activity 12H

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

Area of a Parallelogram
The area of a parallelogram with base b and height h is 𝐴𝑟𝑒𝑎 = 𝑏∙ℎ

Section 12.5: Shearing

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

Examples of Shearing

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

Section 12.6: Area of Circles and the Number π

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 𝐷=2𝑟

The number π Def: The number pi, or π, is the ratio of the circumference and diameter of any circle. That is, 𝜋= 𝐶 𝐷 Circumference Formulas: The circumference of a circle is given by 𝐶=π∙D or 𝐶=2πr

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

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

Area of a Circle The area of a circle with radius 𝑟 is given by 𝐴=π∙ 𝑟 2 See Activity 12O to see why

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?

Section 12.7: Approximating Areas of Irregular Shapes

How do we estimate the area of the following shape?

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

See Example problems in Activity 12Q