Chapter 12: Area of Shapes

Report
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
• 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.
12
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
13
Base and Height Ex’s
14
Area of a Triangle
• The area of a triangle with base b and height h is given by the
formula
1
 = ∙  ∙ ℎ
2
• It doesn’t matter which side you choose as the base!
15
Revisiting Example 1
• Ex 1: Determine the area of
the following triangle.
16
• See problems in Activities 12F and 12G
17
Section 12.4:
Area of Parallelograms
and Other Polygons
• See Activity 12H
19
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
20
Area of a Parallelogram
• The area of a parallelogram with base b and height h is
 =  ∙ ℎ
21
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
23
Examples of Shearing
24
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
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
 = 2
27
The number π
• Def: The number pi, or π, is the ratio of the circumference and
diameter of any circle. That is,

=

• Circumference Formulas: The circumference of a circle is given
by
 = π∙D or  = 2πr
28
Quick Example Problem
• Ex 1: A circular racetrack with a radius of 4 miles has what
length for each lap?
29
How to demonstrate the size of π
• See activities 12M and 12N
30
Area of a Circle
• The area of a circle with radius  is given by
 = π∙ 2
• See Activity 12O to see why
31
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 1 4 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?
32
Section 12.7: Approximating
Areas of Irregular Shapes
How do we estimate the area of the
following shape?
34
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
See Example problems in Activity 12Q
36

similar documents