Area: Parallelograms

1
Area Parallelograms
Lesson 10-1
Objectives 1. to find areas of rectangles
2. to find areas of
parallelograms
2
Area Parallelograms
Lesson 10-1
New Terms 1. area the area of a figure is
the number of square units it encloses
2. altitude an altitude is a
line segment perpendicular to the line
containing a base of the figure and drawn from
the side opposite that base
Tips look for two segments that form a right
angle when determining a base and height
3
Area Parallelograms
Lesson 10-1
Find the area of the rectangle.
Step 1 Change the units so that they are the
same. 150 cm 1.5 m Change 150 centimeters
to meters.
Step 2 Find the area.
A bh Use the formula for area of a
rectangle.
(4)(1.5) Replace b and h with the
dimensions 4 and 1.5.
6 Simplify.
The area of the rectangle is 6 m2.
4
Area Parallelograms
Lesson 10-1
Find the area of each parallelogram.
a.
b.
The area is 16 m2.
The area is 15 in.2.
5
Area Triangles and Trapezoids
Lesson 10-2
Objectives 1. to find areas of triangles
2. to find areas of trapezoids
6
Area Triangles and Trapezoids
Lesson 10-2
New Terms 1. altitude of a triangle is the
perpendicular segment from a vertex of a
triangle to the line containing the opposite
side. The height is the length of the altitude.
Tips bases of a trapezoid are the two sides
parallel to each other, even if the figure is
turned to find the area of an irregular figure,
you may need to use more than one area formula.
(make sure to know all the area formulas)
7
Area Triangles and Trapezoids
Lesson 10-2
Find the area of the triangle.
39 Simplify.
The area is 39 in.2.
8
Area Triangles and Trapezoids
Lesson 10-2
Find the area of the figure.
Add to find the total 450 1,350 1,800.
The area of the figure is 1,800 cm2.
9
Area Triangles and Trapezoids
Lesson 10-2
Suppose that, through the years, a layer of silt
and mud settled in the bottom of the Erie Canal.
Below is the resulting cross section of the
canal. Find the area of the trapezoidal cross
section.
106.5
The area of the cross section is 106.5 ft2.
10
Area Circles
Lesson 10-3
Objectives 1. to find areas of circles
2. to find areas of irregular
figures that include parts of a circle
11
Area Circles
Lesson 10-3
New Terms 1. circumference is the
measurement around the outside of the circle.
2 times the radius times pi 2pr or pd.
2. radius is half the
diameter. 3. diameter
segment that goes from one point of the circle to
another point, and goes through, contains, the
center.
Tips remember to follow the order of operations
when finding the area of a circle or
semi-circle. Square the radius first, and then
multiply by p. when asked to find the exact
area of a circle (or circular shape) do not
substitute 3.14 in for p, leave p as p. make
sure not to confuse the radius as the
diameterknow the difference.
12
Area Circles
Lesson 10-3
Find the exact area of a circle with diameter 20
in.
13
Area Circles
Lesson 10-3
A TV stations weather radar can detect
precipitation in a circular region having a
diameter of 100 mi. Find the area of the region.
7,850 approximate area
The area of the region is about 7,850 mi2.
14
Area Circles
Lesson 10-3
A pound of grass seed covers approximately 675
ft2. Find the area of the lawn below. Then find
the number of bags of grass seed you need to buy
to cover the lawn. Grass seed comes in 3-lb bags.
15
Area Circles
Lesson 10-3
(continued)
Area of region that is a rectangle area of
rectangle bh A 45 25 Replace b with 45 and
h with 25. 1,125
The area of the lawn is about 177 ft2 1,125 ft2
1,302 ft2.
You need to buy one 3-lb bag of grass seed.
16
Space Figures
Lesson 10-4
Objectives 1. to identify common space
figures 2. to identify
space figures from nets
New Terms 1. space figure are
three-dimensional figures, solids net is a
pattern (2-D) you can form into a space figure
Tips a cube is a rectangular prism with six
congruent square faces Lateral means on the
side. The lateral faces of a prism or pyramid
are the surfaces that connect with a base.
17
Space Figures
Lesson 10-4
A prism has two parallel bases that are congruent
polygons, and lateral faces that are
parallelograms
A pyramid has a base that is a polygon. The
lateral faces are triangles.
18
Space Figures
Lesson 10-4
A cylinder has two parallel bases that are
congruent circles.
A cone has one circular base and one vertex.
19
Space Figures
Lesson 10-4
A sphere is the set of all points in space that
are a given distance from a given point called
the center.
20
Space Figures
Lesson 10-4
Describe the bases and name the figure.
The bases are circles.
The figure is a cylinder.
21
Space Figures
Lesson 10-4
Name the space figure you can form from the net.
With two hexagonal bases and rectangular sides,
you can form a hexagonal prism.
22
Surface Area Prisms and Cylinders
Lesson 10-5
Objectives 1. to find surface areas of
prisms 2. to find surface
areas of cylinders
23
Surface Area Prisms and Cylinders
Lesson 10-5
New Terms 1. Surface Area (SA) is the sum
of the areas of the base(s) and the lateral
faces of a space figure. Lateral Area (LA)
is the sum of the areas of the lateral faces.
Tips SA surface Area, LA lateral area, B
area of the base, p perimeter of the base One
way to find the surface area of a space figure is
to find the area of its net. Surface Area is
measured in square units. SA of a cylinder can
also be written as SA 2prh 2pr2
24
Surface Area Prisms and Cylinders
Lesson 10-5
Find the surface area of the rectangular prism
using a net.
Draw and label a net.
60 60 150 90 150 90 600 Add the
areas.
The surface area is 600 cm2.
25
Surface Area Prisms and Cylinders
Lesson 10-5
Find the surface area of the rectangular prism.
Step 2 Find the surface area. S.A. L.A. 2B
Use the formula for surface area.
440 2(5 6) L.A. 440 and B 5
6 440 60 500
The surface area of the rectangular prism is 500
in.2.
26
Surface Area Prisms and Cylinders
Lesson 10-5
Find the surface area of the cylindrical water
tank.
The surface area of the water tank is about 1,156
ft2.
27
Surface Area Pyramids, Cones, and Spheres
Lesson 10-6
Objectives 1. to find surface areas of
pyramids 2. To find surface areas of cones and
spheres
28
Surface Area Pyramids, Cones, and Spheres
Lesson 10-6
New Terms 1. slant height (l ) is the height
of a face, used to find the area of the lateral
faces.
Tips the slant height is not perpendicular to
the base of a pyramid or cone. In a pyramid, it
is perpendicular to the base of a triangular
face. In a cone, it is the shortest segment
that joins the vertex to a point on the circle
base there are a number of analogies that one
can use to remember not only the lateral-area
formulas, but also the surface-area formulas,
each of which involves adding lateral area and
base area(s).
29
Surface Area Pyramids, Cones, and Spheres
Lesson 10-6
Find the surface area of the square pyramid.
Step 2 S.A. L.A. B Use the formula
for surface area.
80 52 Lateral area 80 and B 52.
80 25 105
The surface area of the pyramid is 105 m2.
30
Surface Area Pyramids, Cones, and Spheres
Lesson 10-6
Find the surface area of the cone.
The surface area of the cone is about 94 m2.
31
Surface Area Pyramids, Cones, and Spheres
Lesson 10-6
Earth has an average radius of 3,963 mi. What is
Earths approximate surface area to the nearest
1,000 mi2? Assume that Earth is a sphere.
197,259,434.64 Multiply.
The surface area of Earth is about 197,259,000
mi2.
32
Volume Prisms and Cylinders
Lesson 10-7
Objectives 1. to find volumes of prisms
2. to find volumes of cylinders
33
Volume Prisms and Cylinders
Lesson 10-7
New Terms 1. Volume (V) is the number of
cubic units needed to fill a 3- dimensional
figure. cubic unit is the space occupied by
a cube with edges 1 unit long. (ie. in3)
Tips read the question carefully, you can
calculate both the volume and surface area of a
3-D figure, make sure to understand the concepts
of both and use the correct formula.
34
Volume Prisms and Cylinders
Lesson 10-7
Find the volume of the triangular prism.
V Bh Use the formula for volume.
1,260 Simplify.
The volume is 1,260 cm3.
35
Volume Prisms and Cylinders
Lesson 10-7
Find the volume of the juice can, to the nearest
cubic centimeter.
V Bh Use the formula for volume.
580.7744 Simplify.
The volume is about 581 cm3.
36
Volume Pyramids, Cones, and Spheres
Lesson 10-9
Objectives 1. to find volumes of pyramids and
cones 2. to find volumes of
spheres
37
Volume Pyramids, Cones, and Spheres
Lesson 10-9
Tips the formula for the volumes of cones and
pyramids uses each figures height, not slant
height. Remember, the height of a cone or
pyramid is the length of the segment from the
vertex perpendicular to the base. the volume
of a sphere is the only one (studied so far) that
involves a third power.
38
Volume Pyramids, Cones, and Spheres
Lesson 10-9
Find the volume of the cone.
50.24 Simplify.
The volume of the cone is about 50 in.3.
39
Volume Pyramids, Cones, and Spheres
Lesson 10-9
Find the volume of the square pyramid.
256 Simplify.
The volume of the pyramid is 256 in.3.
40
Volume Pyramids, Cones, and Spheres
Lesson 10-9
Additional Examples
Earth has an average radius of 3,963 mi. What is
Earths approximate volume to the nearest
1,000,000 mi3? Assume that Earth is a sphere.
The volume of the Earth is about 260,580,000,000
mi3.