Chapter 6: Perimeter, Area, and...

transcript

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 +

P = 92 feet

Try this one on your own…

P = S + S + S + S

P = 17.5x + 11x + 17.5x + 11x

P = 57X units

Find the perimeter

of each figure.

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 = 4 X 5

A = 20 units squared

Graph each figure with

the given vertices.

Then find the area of

the figure.

(-4, 0), (2, 0), (4, 3), (-2, 3)

A = bH

A = 6 x 3

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

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.(-1,-3), (0,2), (3,2), (3, -3)

A = ½ x h x (b1 + b2)

A = ½ x 5 x (4 +3)

A = ½ x 5 x (7)

A = 2.5 x 7

A =17.5 units squared

A = ½ x h x (B1 + B2)

A = ½ x 3 x (3 + 5)

A = ½ x 3 x (8)

A = 1.5 x 8

Graph and find the

area of each figure

with the given

vertices. (-3,-2), (-3,1), (0,1), (2, -2)

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.

Find the length of

the hypotenuse.

C = 6.40

Graph the triangle

with the following

coordinates (1,-2),

(1,7), and (13,-2).

B = 12

Find the length of

the hypotenuse.

C = 15

Example 2: Finding the length of a

Leg in a Right Triangle

Solve for the

unknown side in

the right triangle.

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.

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

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

Step One: Graph Circle

Step Two: Find the radius

Step Three: Use the Area and

Circumference Formula

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

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 Revolutions88/21 x 147 = 616 feet

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.� 2 units by 4 units

� 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

� 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

Step Two: Plug the

numbers into the

formula.

Step Three: Solve

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.

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

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

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

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?

Composite Figures

Find the volume of

the milk carton.

Find the volume of the barn.

Section 6.7: Volume of Pyramids

and Cones

Pyramids and Cones

Find the volume of

each figure.

Try this one on

your own…

Find the volume of

each figure.

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

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

Pyramids and Cones

Find the volume of

each figure.

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

Try this one on

your own…

Try this one on

your own…

Find the surface

Finding the surface

Try this one on

your own…

Finding the surface

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

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.

Original Triple Radius Triple Height

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

Find the surface

Try this one on

your own…

Find the surface

Try this one on

your own…

Find the surface

Try this one on

your own…

Find the surface

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

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.

Original Triple Radius Triple Slant Height

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?

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.

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

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

Chapter 6: Perimeter, Area, and...

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