Chapter 10

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Transcript Chapter 10

Chapter 10
Measurement
Section 10.2
Perimeter and Area
Measurements of Two-Dimensional Shapes
Two-Dimensional shapes have many features to them. There are two of them in
this section that we discuss measuring because it is useful to be able to calculate
for many different reasons.
Perimeter
The perimeter of a simple closed plane figure is the length of its boundary. The
difficulty is that the boundary might be made out of several segments or curves
that need to be measured individually and added together.
Perimeter of Polygons
To find the perimeter of a polygon measure the lengths of the sides and add them
2
together.
4
2
2
5
5
2
6
6
4
2
7
7
5
1
3
Perimeter =
Perimeter =
Perimeter =
Perimeter =
3+ 4+5 = 12
2+5+2+5 = 14
4+6+1+6 = 17
2+2+2+7+7=20
On a geoboard a unit is usually the horizontal or vertical distance between two
consecutive dots. I can be other things but this is what is considered to be
standard when no other unit is mentioned.
2
4
3
1
3
Find the
perimeters of
the two shapes
pictured here.
1
6
4
5
4
7
6
2 + 3 + 3 + 1 + 1 + 4 + 6 + 6 = 26
4 + 5 + 7 + 4 = 20
Circumference: Perimeter of a Circle
The perimeter of a circle (or the distance around the outside is
called the circumference. It was discovered long ago the ratio
between the circumference (C) and the diameter (d) (or twice
the radius (2r)) is the number  (pi). We get the formulas:
22
 
 3.14
7
C

d
or
C

2r
or
C  d
or
C  2r
1
44=
22=
4
4
3
8
The length of a part of the circumference is split up
just like the circle. If the circle is cut in half so is the
circumference. If the circle is cut in quarters so is
the circumference. Find the perimeter of the shape
to the left.
 + 3 +  + 1 + 4 + 8 + 4 = 20 + 2
Area
The area of a two dimensional shape is a measure of how much space it takes up.
This has very practical uses such as determining the amount of paint needed to
paint a room or the amount of carpet needed to cover a floor.
Area is usually measured in square units ( i.e. square inches, square feet, square
meters etc.) although shapes other than squares can be used, but squares are the
most common. The area is the number of non-overlapping squares that are
required to cover up the shape. Here are some examples.
1
2
3 4
5 6
7
8
9
10
11
12
13
14
15
16
17
18
Area = 18 square green units
1
2
3
4
5
6
7
8
Area = 8 square yellow units
Area & Perimeter of Rectangles
A rectangle is a shape for which the area and perimeter can be found by
measuring just two distances we call the length (l) and width (w). A formula is
given for each of the area and perimeter below.
l
Area = (length)·(width) = lw
w
Perimeter = 2·(length) + 2·(width) = 2l + 2w
2
1
3
1
3
6
Area =
A unit square on a geoboard is usually taken to be a
square going over 1 unit and down 1 unit. Finding the
area of a shape on a geoboard can be done by
breaking the shape up into shapes you are more
familiar with (such as rectangles) and computing the
area of those shapes. This method utilizes van Hiele
levels one and two to recognize one shape being
made out of other less complicated shapes.
First break this shape up into 3 rectangles.
2 · 3 + 1 · 1 + 3 · 6 = 6 + 1 + 18 = 25 square units
Another common method for finding the area of a shape
besides cutting it apart is to fill in the missing part and
remove it.
4
4
For example in the shape to the left. Fill in the missing
part to make a rectangle.
The area of the rectangle is 3·4 =12, but the part that
makes up the shape is only half of that which is 122=6.
This gives a total area of: 4·4 + 6 = 22 square units.
What is the area of the rectangle below measured in
blue rectangular units given to the right?
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
The area is 24 blue rectangular units.