Section 9.3 class notes
Perimeter
The distance around a two-dimensional shape.
Example: the perimeter of this rectangle is 3+7+3+7 = 20 units
circumference - the perimeter of a circle is called the
Area
Definition of Area

Area is defined as the number of square units that covers a closed figure.
Example of Area

In the example shown, the area of the square is 16 square units. That means, 16 square units
are needed to cover the surface enclosed by the square.
There are several geometric shapes that you will need to know the area and perimeter formulas for.
Shapes
Formula
Rectangle:
Area = Length X Width
A = lw
Perimeter = 2 X Lengths + 2 X Widths
P = 2l + 2w
Parallelogram
Area = Base X Height
a = bh
Perimeter = 2a + 2b
Triangle
Area = 1/2 of the base X the height
a = 1/2 bh
Perimeter = a + b + c
(add the length of the three sides)
b
Trapezoid
Perimeter
P = a + b1 + b2 + c
Circle
The distance around the circle is a
circumference. The distance across the
circle is the diameter (d). The radius
(r) is the distance from the center to a
point on the circle. (Pi = 3.14)
d = 2r
c = d = 2 r
A = r2
=3.14)
You should also be aware of the Pythagorean Theorem (which only applies to RIGHT Triangles)
a
c
b
a2 + b2 = c2
Here is a quick review of how the Pythagorean Theorem works. The
Pythagorean Theorem only works in a 90 degree triangle. This means that if you know
two of the sides, you can always find the third one.
In the right triangle at the left, we know that:
h2 = 62 + 82 (the h is the c in this triangle)
Simplifying the squares gives:
h2 = 36 + 64
and then:
h2 = 100
h = 10
(by doing the square root of 100)
Here's another one:
In this example, the missing side is not the long one. But the
theorem still works, as long as you start with the hypotenuse:
152 = x2 + 92 (this x is either an a or a b)
Simplifying the squares gives:
225 = x2 + 81
and then:
225 - 81 = x2
144 = x2
12 = x
(Notice that we had to rearrange the equation)
Now something different will happen:
In the right triangle at the left, we know that:
h2 = 72 + 102
Simplifying the squares gives:
h2 = 49 + 100
h2 = 149
This square root is not perfect. A calculator gives:
h = 12.2
(rounded to one decimal place)
And one more example:
Again, the missing side is not the long one. But we start with
the hypotenuse:
182 = x2 + 112
Simplifying the squares gives:
324 = x2 + 121
and then:
324 - 121 = x2
203 = x2
14.2 = x
(We had to rearrange the equation, and round the answer.)
Use the Pythagorean Theorem to find the missing length. Then find the perimeter and area of each figure. Round to
two decimal places when necessary. Make sure to use correct units in your answers.
Find the perimeter and area for each of the following geometric shapes. Give correct units in your answer. Use the
π button on your calculator when needed and round to 2 decimal places.
5 ft.
1.
2.
2 cm
3 ft
2 ft
Solution:
Solution:
Perimeter =
Perimeter =
Area A =
Area A =
3.
4.
13 in.
10 m
8m
7m
6m
5 in.
Solution:
Solution:
Perimeter =
Area
Perimeter =
Area =
A=
8m
5.
4 yd
5 yd
Solution:
Perimeter =.
Area =
Find the area of the “shaded” region in the diagrams below. Be sure to give appropriate units in your answers!
6.
radius of circle = 4 inches
Solution:
7.
Solution:
The radius of the “outer” circle is 10 inches. The inner circle has a radius that is
4 inches.
Here are a few problems from the section.
One square yard equals 9 square feet. Use this information to convert the following.
Problem 34 from section 9.3 - 15.2 ft2 to square yards
Problem 36 from section 9.3 - 18.3 yd2 to square feet
We need to do a few of the word problems in section. We will need to use the diagrams on page 505 to answer
these questions.
Section 9.4 class notes
II. Volume and Surface area.
Volume
The amount of 3-dimensional space an object occupies. Capacity..
For this example the volume is 4×5×10 = 200 units3
Units of volume include:
Metric: cubic centimeters (cm3), cubic meters (m3), liters
Surface Area
The total area of the surface
of a three-dimensional object.
It is found by adding the area
of each face of the object.
Here are some volume formulas and surface area formulas that we will need..
Rectangular Solid
Volume = Length X Width X Height
V = lwh
Surface = 2lw + 2lh + 2wh
Prisms
Volume = Base X Height
v=bh
Surface = 2b + Ph (b is the area of the
base P is the perimeter of the base)
Cylinder
Volume = r2 x height
V = r2 h
Surface = 2 radius x height
S = 2rh + 2r2
Pyramid
V = 1/3 bh
b is the area of the base
Surface Area: Add the area of the base to
the sum of the areas of all of the
triangular faces. The areas of the
triangular faces will have different
formulas for different shaped bases.
Cones
Volume = 1/3 r2 x height
V= 1/3 r2h
Surface = r2 + rs
S = r2 + rs
=r2+r
Sphere
Volume = 4/3 r3
V = 4/3 r3
Surface = 4r2
S = 4r2
Examples: Find the volume and surface area of the following geometrical solids. Be sure to include proper units
with your answer.
8.
6 cm
9.
5cm
10 ft
3 ft
2 ft
Solution:
Volume =
Solution:
Volume =
Surface Area
S=
Surface Area
S=
9.
8m
10. This Sphere has radius 4 miles.
25 m
Solution:
Volume V= 1/3 r2h 
Surface Area = r2+r



Solution:
Volume V = 4/3 r3
Surface Area = 4r2
We should try a few problems from the text that I just can’t draw.
Try 22, 24 and 26 in section 9.4.
Here are a few more problems from the text.
Exercises 29 – 32, use the fact that 1 yd3 equals 27 ft3 to make the conversion
Problem 30 from section 9.4 - 3.8 yd3 to cubic feet.
Problem 32 - 2457 ft3 to cubic yards.
39. A bread pan is 12 in X 4 in X 3 in. How many quarts does it hold if 1 in 3 = 0.01736 qt.
42a) what is the volume of water in a rectangular swimming pool that is 15 m long and 9 m
wide and has an average depth of 2 m? Give your answer in cubic meters.
42b) If 1 m3 = 1 kl, how many kiloliters of water will it hold?