Chapter 6: Perimeter, Area,
and Volume
Regular Math
Section 6.1: Perimeter & Area of
Rectangles & Parallelograms
Perimeter – the distance around the
OUTSIDE of a figure
Area – the number of square units
INSIDE a figure
Finding the Perimeter of Rectangles
and Parallelograms
Find the perimeter
of each figure.
P=S+S+S+S
P = 26 + 20 + 26 +
20
P = 92 feet
Try this one on your own…
Find the perimeter
of each figure.
P=S+S+S+S
P = 17.5x + 11x + 17.5x + 11x
P = 57X units
Using a Graph to Find Area
Graph each figure with the
given vertices. Then find the
area of each figure.
(-3, -1), (-3, 4), (1, 4), (1, -1)
A = bH
b = base ; H =
height
A=4X5
A = 20 units squared
Try this one on your own…
Graph each figure with
the given vertices.
Then find the area of
the figure.
A = bH
A=6x3
(-4, 0), (2, 0), (4, 3), (-2, 3)
A = 18 units squared
Finding Area and Perimeter of a
Composite Figure
Step One: Fill in the
missing sides.
Step Two: Solve for
Perimeter
Step Three: Break the
figure into rectangles.
Step Four: Solve for Area
of each rectangle.
Step Five: Add the areas of
each individual rectangles.
Find the perimeter and
area of the figure.
Section 6.2: Perimeter and Area of
Triangles and Trapezoids
Find the perimeter
of each figure.
P=S+S+S
P = 22 + 22 + 27
P = 71 feet
Try this one on your own…
Find the perimeter
of each figure.
P=S+S+S
P = 2.5x + 5y + 2x + 2x + 4y
P = 6.5x + 9y
Find the area of triangles and
trapezoids.
Graph and find the
area of each figure
with the given
vertices.
A = ½ x h x (b1 + b2)
(-1,-3), (0,2), (3,2), (3, -3)
A = ½ x 5 x (7)
A = ½ x 5 x (4 +3)
A = 2.5 x 7
A =17.5 units squared
Try this one on your own…
A = ½ x h x (B1 + B2)
A = ½ x 3 x (3 + 5)
Graph and find the
area of each figure
with the given
vertices.
(-3,-2), (-3,1), (0,1), (2, -2)
A = ½ x 3 x (8)
A = 1.5 x 8
A = 12 units squared
Section 6.3: The Pythagorean
Theorem
Example 1: Finding the length of
the hypotenuse.
Find the length of
the hypotenuse.
Graph the triangle
with coordinates
(6,1), (0,9), and
(0,1).
Find the length of
the hypotenuse.
Try this one on your own…
Find the length of
the hypotenuse.
Graph the triangle
with the following
coordinates (1,-2),
(1,7), and (13,-2).
A=9
B = 12
Find the length of
the hypotenuse.
C = 15
C = 6.40
Example 2: Finding the length of a
Leg in a Right Triangle
Solve for the
unknown side in
the right triangle.
Try this one on your own…
Solve for the
unknown side in
the right triangle.
b = 24
Example 3: Using the Pythagorean
Theorem to Find Area
Use the
Pythagorean
Theorem to find the
height of the
triangle.
Then, use the
height to find the
area of the triangle.
Try this one on your own…
Use the
Pythagorean
Theorem to find the
height of the triangle.
h = square root of 20
or 4.47
Then, use the height
to find the area of
the triangle.
A = 17.89 units
squared
Section 6.4: Circles
Finding the circumference of a
Circle.
Find the circumference of each circle,
both in terms of pi and to the nearest
tenth. Use 3.14 for pi.
Circle with radius 5 cm
Circle with diameter 1.5 in
Try these on your own…
Find the
circumference of
each circle, both in
terms of pi and to
the nearest tenth.
Use 3.14 for pi.
Circle with radius 4 m
C = 8pi m or 25.1 m
Circle with
diameter 3.3 ft
C = 3.3pi or 10.4 ft
Finding the Area of a Circle.
Find the area of each circle, both in
terms of pi and to the nearest tenth.
Use 3.14 for pi.
Circle with radius 5 cm
Circle with diameter 1.5 in
Try these on your own…
Find the area of
each circle, both in
terms of pi and to
the nearest tenth.
Use 3.14 for pi.
Circle with radius 4
in
A = 16pi inches
squared or 50.2
inches squared
Circle with diameter
3.3 m
A = 2.7225pi meters
squared or 8.5
meters squared
Finding Area and Circumference on
a Coordinate Plane.
Graph the circle with center (-1,1) that
passes through (-1,3). Find the area
and circumference, both in terms of pi
and to the nearest tenth. Use 3.14 for
pi.
Step One: Graph Circle
Step Two: Find the radius
Step Three: Use the Area and
Circumference Formula
Try this one on your own…
Graph the circle with center (-2,1) that
passes through (1,-1). Find the area
and circumference, both in terms of pi
and to the nearest tenth. Use 3.14 for
pi.
A = 9pi units squared and 28.3 units
squared
C = 6pi units and 18.8 units
A bicycle odometer recorded 147
revolutions of a wheel with diameter 4/3
ft. How far did the bicycle travel? Use
22/7 for pi.
The distance traveled is the
circumference of the wheel times the
number of revolutions.
C = pi(d) = (22/7) (4/3) = 88/21
Circumference x Revolutions
88/21 x 147 = 616 feet
Try this one on your own…
A Ferris wheel has a diameter of 56
feet and makes 15 revolutions per
ride. How far would someone travel
during a ride? Use 22/7 for pi.
C = 22/7(56) = 176 feet
Distance = 176 (15) = 2640 feet
Section 6.5: Drawing Three
Dimensional Figures
Example 1: Drawing a Rectangular Box
 Use isometric dot paper to sketch a rectangular box
that is 4 units long, 2 units wide, and 3 units high.

Step 1: Lightly draw the edges of the bottom face. It will
look like a parallelogram.


Step 2: Lightly draw the vertical line segments from the
vertices of the base.


3 units high
Step 3: Lightly draw the top face by connecting the
vertical lines to form a parallelogram.


2 units by 4 units
2 units by 4 units
Step 4: Darken the lines.

Use solid lines for the edges that are visible and dashed lines
for the edges that are hidden.
Example 2: Sketching a One-Point
Perspective Drawing
Step 1: Draw a rectangle.
This will be the front face.
Label the vertices A through D.
Step 2: Mark a vanishing point “V” somewhere above
your rectangle, and draw a dashed line from each
vertex to “V”.
Step 3: Choose a point “G” on line BV. Lightly draw a
smaller rectangle that has G as one of its vertices.
Step 4: Connect the vertices of the two rectangles
along the dashed lines.
Step 5: Darken the visible edges, and draw dashed
segments for the hidden edges. Erase the vanishing
point and all the lines connecting it to the vertices.
Example 3: Sketching a Two-Point
Perspective Drawing
Step 1: Draw a vertical segment and label it AD.
Draw a horizontal line above segment AD. Label
vanishing points V and W on the line. Draw
dashed segments AV, AW, DV, and DW.
Step 2: Label point C on segment DV and point
E on segment DW. Draw vertical segments
through C and E. Draw segment EV and CW.
Step 3: Darken the visible edges. Erase horizon
lines and dashed segments.
Section 6.6: Volume of Prisms
and Cylinders
Example 1: Finding the Volume of
Prisms and Cylinders
Find the volume of
each figure to the
nearest tenth.
Step One: Figure
out what formula to
use.
Step Two: Plug the
numbers into the
formula.
Step Three: Solve
Try this one on your own…
Find the volume of
each figure to the
nearest tenth.
Example 2: Exploring the Effects of
Changing Dimensions
A juice can has a radius of 1.5 inches and
a height of 5 inches. Explain whether
doubling the height of the can would have
the same effect on the volume as doubling
the radius.
Original
Double
Radius
Double Height
Try this one on your own..
A juice can has a radius of 2 inches
and a height of 5 inches. Explain
whether tripling the height would have
the same effect on the volume as
tripling the radius.
Example 1: Finding the Volume of
Prisms and Cylinders
Find the volume of
each figure to the
nearest tenth.
A rectangular prism
with base 1 meter
by 3 meters and
height of 6 meters
Try these on your own…
Find the volume of
each figure to the
nearest tenth.
A rectangular prism
with base 2 cm by
5 cm and a height
of 3cm
Example 2: Exploring the Effects of
Changing Dimensions
A juice box measures 3 inches by 2 inches
by 4 inches. Explain whether doubling the
length, width, or height of the box would
double the amount of juice the box holds.
Original
Length
Width
Height
Try this one on your own…
A juice box measures 3 inches by 2 inches
by 4 inches. Explain whether tripling the
length, width, or height would triple the
amount of juice the box holds.
Original
Length
Width
Height
Example 3: Construction
Application
Kansai International
Airport is a man-made
island that is a
rectangular prism
measuring 60 ft deep,
4000 ft wide, and 2.5
miles long. What is the
volume of rock, gravel,
and concrete that was
needed to build the
island?
Try this one on your
own…
A section of an airport
runway is a rectangular
prism measuring 2 feet
thick, 100 feet wide,
and 1.5 miles long.
What is the volume of
material that was
needed to build the
runway?
Example 4: Finding the Volume of
Composite Figures
Find the volume of
the milk carton.
Try this one on your own…
Find the volume of the barn.
Section 6.7: Volume of Pyramids
and Cones
Example 1: Finding the Volume of
Pyramids and Cones
Find the volume of
each figure.
Try this one on
your own…
Find the volume of
each figure.
Example 2: Exploring the Effects of
Changing Dimensions
A cone has a radius 7 feet and height 14
feet. Explain whether tripling the height
would have the same effect on the volume
of the cone as tripling the radius.
Original
Triple Height
Triple Radius
Try this one on your own…
A cone has a radius 3 feet and height 4
feet. Explain whether doubling the height
would have the same effect on the volume
as doubling the radius.
Original
Double Height
Double Radius
Example 1: Finding the Volume of
Pyramids and Cones
Find the volume of
each figure.
Try these on your own…
Find the volume of
each figure.
Example 3: Social Studies
Application
The Great Pyramid
of Giza is a square
pyramid. Its height
is 481 feet, and its
base has 756 feet
sides. Find the
volume of the
pyramid.
Try these on your
own…
The pyramid of
Kukulcan in Mexico
is a square
pyramid. Its height
is 24 meters and its
base has 55 meter
sides. Find the
volume of the
pyramid.
Section 6.8: Surface Area of Prisms
and Cylinders
Example 1: Finding Surface Area
Find the surface
area of each figure.
Try this one on
your own…
Try this one on
your own…
Find the surface
area of each figure.
Find the surface
area of each figure.
Example 1: Finding Surface Area
Finding the surface
area of each figure.
Try this one on
your own…
Finding the surface
area of each figure.
Example 2: Exploring the Effects of
Changing Dimensions
A cylinder has a diameter of 8 inches and a
height of 3 inches. Explain whether
doubling the height would have the same
effect on the surface area as doubling the
radius.
Original
Double Height
Double Radius
Try this one on your own…
Original
Triple Radius
Triple Height
A cylinder has a diameter of 8 inches and a
height of 3 inches. Explain whether tripling
the height would have the same effect on
the surface area as tripling the radius.
Example 3: Art Application
A web site advertises
that it can turn your
photo into an
anamorphic image. To
reflect the picture, you
need to cover a
cylinder that is 32mm
in diameter and 100
mm tall with reflective
material. How much
reflective material do
you need?
Try this one on your
own…
A cylindrical soup can
has a radius of 7.6 cm
and is 11.2 cm tall.
What is the area of the
label that covers the
side of the can?
Section 6.9: Surface Area of
Pyramids and Cones
Example 1: Finding Surface Area
Find the surface
area of each figure.
Try this one on
your own…
Find the surface
area of each figure.
Try this one on
your own…
Find the surface
area of each figure.
Find the surface
area of each figure.
Example 1: Finding Surface Area
Try this one on
your own…
Find the surface
area of each figure.
Find the surface
area of each figure.
Example 2: Exploring the Effects of
Changing Dimensions
A cone has a diameter 8 in. and slant
height 5 in. Explain whether doubling the
slant height would have the same effect on
the surface area as doubling the radius.
Original
Double Slant Height
Double Radius
Try this one on your own…
Original
Triple Radius
Triple Slant Height
A cone has diameter of 8 in. and slant
height 3 in. Explain whether tripling the
slant height would have the same effect on
the surface area as tripling the radius.
Example 3: Life Science Application
An ant lion pit is an
inverted cone with
the dimensions
shown. What is the
lateral surface area
of the pit?
Try this one on your own…
The upper portion
of an hourglass is
approximately an
inverted cone with
the given
dimensions. What
is the lateral
surface area of the
upper portion of
the hourglass?
Section 6.10: Spheres
Example 1: Finding the Volume of a
Sphere
Find the volume of a sphere with a radius
of 6 ft, both in terms of pi and to the
nearest tenth.
Try this one on your own…
Find the volume of a sphere with radius 9 cm,
both in terms of pi and to the nearest tenth.
Example 2: Finding Surface Area of
a Sphere
Find the surface
area, both in terms
of pi and to the
nearest tenth.
Try this one on
your own…
Find the surface
area, both in terms
of pi and to the
nearest tenth.
Example 3: Comparing Volumes
and Surface Areas
Compare the volume and surface
area of a sphere with radius 21 cm
with that of a rectangular prism
measuring 28 x 33 x 42cm.
Sphere –
Volume
Sphere –
Surface Area
Prism –
Volume
Sphere –
Surface Area
Try this one on your own…
Compare volumes and surface areas
of a sphere with radius 42 cm and a
rectangular prism measuring 44 cm
by 84 cm by 84 cm.
Sphere –
Volume
Sphere –
Surface Area
Prism –
Volume
Prism –
Surface Area