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.
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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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Base and Height Ex’s
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Area of a Triangle
• The area of a triangle with base b and height h is given by the
formula
1
𝐴𝑟𝑒𝑎 = ∙ 𝑏 ∙ ℎ
2
• It doesn’t matter which side you choose as the base!
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Revisiting Example 1
• Ex 1: Determine the area of
the following triangle.
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• See problems in Activities 12F and 12G
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Section 12.4:
Area of Parallelograms
and Other Polygons
• See Activity 12H
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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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Area of a Parallelogram
• The area of a parallelogram with base b and height h is
𝐴𝑟𝑒𝑎 = 𝑏 ∙ ℎ
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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
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Examples of Shearing
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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
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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
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The number π
• Def: The number pi, or π, is the ratio of the circumference and
diameter of any circle. That is,
𝐶
𝜋=
𝐷
• Circumference Formulas: The circumference of a circle is given
by
𝐶 = π∙D or 𝐶 = 2πr
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Quick Example Problem
• Ex 1: A circular racetrack with a radius of 4 miles has what
length for each lap?
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How to demonstrate the size of π
• See activities 12M and 12N
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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
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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?
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Section 12.7: Approximating
Areas of Irregular Shapes
How do we estimate the area of the
following shape?
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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
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See Example problems in Activity 12Q
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