Transcript Area and Circumference

Areas of Polygons
& Circumference of Circles
Define: Area
Area has been defined* as the following:
“a two dimensional space measured by the number
of non-overlapping unit squares or parts of unit
squares that can fit into the space”
Discuss...

*State of Arizona 2008 Standards Glossary
GEOBOARDS
Geoboards are wonderful tools for exploring the concept of area. To change the
picture all we have to do is move the geobands! If you do not have geoboards, use
the grids to do the explorations.
Start by making as many different sized squares as you can on your geoboard.
Sketch them on the grids below.
What is the area of the smallest square?
What is the area of the largest square?
Make as many different rectangles as you can that have an area of 4 square units.
Sketch them on the grids. Find their perimeters. Are the perimeters all the same?
GEOBOARDS
Make as many different rectangles as you can that have an area
of 4 square units.
Sketch them on the grids. Find their perimeters. Are the
perimeters all the same?
Finding Triangles:
Make as many different triangles as you can that have an area of
2 square units.
Sketch them on the grids. Explain, in your own words, how
your formed them.
Area
 Find the area of each polygon by counting unit squares.
Areas of Irregular Figures
 Find the area of each of the figures. Make sure to keep track
of your work and/or the process you took as you found the
area.
Areas on a Geoboard
 Addition method
divide an area into
smaller pieces and then
add the areas.
What is the area of
this figure?
19 square units
A
36
complete
squares
36
complete
squares
2
from 4
halves
36
complete
squares
2
from 4
halves
1½
from 1 x 3
triangle
36
complete
squares
2
from 4
halves
1½
from 1 x 3
triangle
3½
from 1 x 7
triangle
36 + 2 + 1.5 + 3.5 = 43
B
Triangle Area
= ½ bh
Base = 5
Height = 10
½ (5)(10) = 25
C.
9 x 8 = 72
TAKE OUT
(4x3)/2 = 6
5x3 = 15
1x4 = 4
(1x1)/2 = 0.5
(3x1)/2 = 1.5
72 – 6-12.5-8-15-4-0.5-1.5 = 24.5
D
18
complete
squares
18
complete
squares
1
from 1 x 2
triangle
18
complete
squares
1
from 1 x 2
triangle
7
from 7 x 2
triangle
18 + 1 + 7
= 26
Rectangle method – construct a rectangle encompassing the
entire figure and then subtract the areas of the unshaded regions.
E
Area = 16 – (3 + 1 + 1 + 1 + 1) = 9
F
Find the area of the figure.
F
The area of the hexagon equals the area of
the surrounding rectangle minus the sum of
the areas of figures a, b, c, d, e, f, and g.
Area and Perimeter Connections
Consider a rectangle that has length and width measurements that are whole numbers.
Given the below conditions determine the length and width measurements for two
examples. If it is not possible to create such a rectangle, explain why.
1. The area is 30 square units.
2. The perimeter is 30 units.
3. The area is 25 square units.
4. The perimeter is 25 units.
5. The area is an even whole number
6. The perimeter is an even whole number.
7. The area is an odd whole number.
8. The perimeter is an odd whole number.
9. The area is a prime number.
10. The perimeter is a prime number.
11. What generalizations can be made regarding area and perimeter of rectangles?
Consider only whole numbers in your generalizations.
12. What generalizations can be made about the relationship between area and perimeter of
a rectangle? Consider only whole numbers in your generalizations.
Finding Area by Dissection
1. How do you compute the area of a rectangle?
Area = length x width
= lw
2. Illustrate a concrete method of finding the
area of a rectangle.
3
6
6 x 3 = 18
 Use the rectangle you created to find the formula
for the area of a triangle.
 If the formula for the area of the triangle is half of
the rectangle, why is the formula ½ bh rather
than ½ lw?
Figure
One
a. Can you make a non-rectangular parallelogram with these two pieces?
b. Describe the process from part a.
A right triangle was cut from one end of the
rectangle and slid to the other side to create a nonrectangular parallelogram.
c. Based on your observation, write a sentence describing the
area of a parallelogram.
The area of the rectangle is equal to the area of the parallelogram.
The width of the rectangle is equal to the height of the
parallelogram and the length is equal to the base.
d. Write a formula for the area of a parallelogram.
Area = base x height
Area = bh
Figure Two
Cut out both figure two shapes from your material
sheet.
a. What are the two shapes? Trapezoids
What word describes the relationship between the
Congruent
two shapes?
b. Put the two shapes together to form a
parallelogram.
c. Describe the process from part b.
Two congruent trapezoids were put
together by rotating one of them 180o to
form a parallelogram
Figure Two continued
d. Based on your observations, write a sentence describing the area
of one of these shapes.
Top
Bottom
Bottom
Top
The area of the trapezoid is half the area of the
parallelogram (½bh). The base of the parallelogram is
equal to the top + bottom of the trapezoid.
Area = ½ (top + bottom) height
Area = ½ (a + b)h = ½ (b1 + b2)h
Problem Solving Application.
You have an unusually shaped pool and you
need to buy a pool cover. The pool cover
cost $4.25 per square foot. How much will it
cost to cover your pool? Triangle: ½ bh =
3 ft 8 ft
8 ft
12 ft
10 ft
4 ft
½ (3 x 8) = 12 ft2
Rectangle: lw =
10 x 8 = 80 ft2
Parallelogram: bh =
12 x 4 = 48 ft2
Pool Area = 12 + 80 + 48 = 140ft2
Cost = 140 x $4.25 = $595
Find the areas:
20 cm2
56 cm2
Area of Polygons
Triangle
Rectangle
Square
Parallelogram
Trapezoid
Odd Shape Polygon
Area of Polygons
Triangle
A 
1
2
bh
Rectangle
A = lw
Square
A = s2
Parallelogram
A = bh
Trapezoid
A = ½ (b1 + b2)h
Odd Shape Polygon
Break into known polygons
How can you use these shapes to come up with
the formula for the area of a: rectangle,
parallelogram, triangle, and parallelogram?
What is a circle?
• Share definitions.
A collection of points equidistant
from a given point
 What does a compass do?
• Draw a circle with your compass.
Circumference
 What is a circumference?
The total distance around a circle.
• What is a diameter?
Diameter explorations
 Take one of the circular objects and piece of string.
 Mark the length of the diameter on your piece of string
 How many of those diameters fit around the circumference
of your circular object?
Understanding Circles
Locate at least 4 round objects and measure the
diameter and measure the circumference of each.
Record your results in the table below. Be sure to
include the units you used in the measuring
process.
Object
measured
Circumference
Diameter
Circumference
÷ Diameter
3.14 = pi ( )
Circumference
Pi = Circumference ÷ diameter
Circumference =  d
Circumference = 2 r
Circumference of a Circle
Circle – the set of all points in a plane that are the same distance
from a given point, the center.
Circumference – the perimeter of a circle.
Pi – the ratio between the
circumference of a circle
and the length of its
diameter.
Find each of the following:
a. The circumference of a circle with radius 2 m. Leave answer in
terms of pi
4π m
b.The radius of a circle with circumference 15π m
7.5 m
Which is the shorter route?
Discovering and Relating Area Formulas
Using Dot Paper
 Area is a spatial concept – a covering of two-dimensional
space.
 Complete the tables
 What do you need to watch for with your students when doing
the triangle, parallelogram, and trapezoid?
 Generalize patterns in the table
 Write the generalized formula for each
Alpha Shapes
 Sort the alpha shapes into two different categories
Capture the Quadrilaterals
 You will need the quadrilaterals from the alphashapes
and a partner!
 Cut on the dotted lines. Make 2 piles of cards: (1) one
pile contains all attributes referring to angles and (2) the
other pile contains all attributes referring to sides. Piles
should be placed upside down. Upon your turn, take one
card from each pile. You then look for all the quadrilateral
alphashapes that match both the attributes on the side
and angle cards. Those quadrilaterals that match both
become “captured” by you. It is than your opponents turn.
He/she will follow the same procedures. Once a
quadrilateral is “captured” it may not be taken unless the
“WILD CARD” is used. The player(s) with the most
“captured” quadrilaterals at the end is the winner!
What is My Shape?
Use one set of Shapes – spread them out in the middle of the table.
Group members take turns being the chooser.
The chooser chooses one of the Shapes while group members need to find the shape that
matches the shape the chooser choose.
Group members are only allowed to ask the chooser “yes” or “no” questions to help
narrow down the possibilities.
Group members are not allowed to point to a shape and ask, “Is this the one?”
Also, group members are not allowed to ask questions about the letter on the shape.
Rather, they must continue to ask questions that reduce the choices to one shape by using
different attributes of the Shapes.
Once a group has an idea what shape was chosen, they ask the chooser if they are
correct.
Alpha Shapes and Area
 Using the centimeter grid paper, find the area of shape W,
you may need to estimate.
 Comparing Shapes – Two V’s
 What is the area of V?
 How can you use V to help find the area of C?
Comparing Shapes
1) Compare the area of each piece to the areas of W and Q.
Sort the shapes into five piles, as shown:
a) Area less than W
b) Area same as W
c) Area greater than W but less than Q
d) Area same as Q
e) Area greater than Q
2) Write about your results and explain how you compared
areas.
Class Discussion
 “How did you decide what strategy to use to find the area of
each shape?”
 “Which area measurements are rough estimates? Which ones
are exact? Why?”
 “Did you find shapes that are not congruent that have the
same area?”
Good Questions
 Each group will work through 3 - 5 of the problems to come
up with an answer key for the entire set of problems
 *Questions from Good Questions for Math Teaching Grades
5 – 8 and Good Questions for Math Teaching Grades K – 6.
Who Put the Tan in Tangram?
 Find the area of each of the shapes from a set of tangrams
 Continue working through the packet from Georgia
Department of Education.