Transcript Geometry

Geometry
Geometry: Part IV
Area and Volume
By
Dick Gill, Julia Arnold and
Marcia Tharp
for
Elementary Algebra Math 03 online
Area
Now that you have completed your module on Perimeter we will
turn our attention to Area. Perimeter is the distance around the
outside of a figure while the area of the figure is the space contained
within the figure. The sketch below for example, represents a square
that measures 1 ft. on each side. Its perimeter would be 4 ft. Its area
would be 1 square foot or 1 sq. ft. or 1 ft2.
1 ft.
1 ft.
The perimeter is the
distance around the
outside: 4 feet or 4 ft.
The area is the space contained inside the
figure. Here, the area is 1 square foot or 1 sq.ft
or 1 ft2.
If we put two of the squares together, we have a rectangle with a
width of 1 ft., a length of 2 ft. and a area of two square feet.
2 ft.
1 ft.
If we put six of the
squares together as
shown in the next
sketch, we have a
rectangle with a width
of 2 ft and a length of 3
ft. and an area of 6 ft2.
3 ft.
2 ft.
2 ft
Here is another rectangle. What is the
perimeter? What is the area?
By definition, the perimeter is the
distance around the outside of the
figure.
4 ft
2 ft + 4 ft + 2 ft + 4 ft = 12 ft
By formula,
P = 2L + 2W = 2(4 ft) + 2(2 ft) = 12 ft.
For the area, you can count 8 sq.ft.
What would be a formula for the area?
Are you ready to predict the formula for the area of a rectangle?
Think before you click.
Just multiply the length times the width and you have the area
of the rectangle. If the length and width are in meters, the area
will be in square meters. If the length and width are in miles,
the area will be in square miles. If the length and width are in
centimeters, the area will be in square centimeters.
See if you can find the area of each of the following rectangles.
The first rectangle has dimensions: L = 5 cm and W = 3 cm.
5 cm
A = LW
3 cm
A = (5 cm)(3 cm)
A = 15 cm2 or
A = 15 sq cm
OK, how about a rectangle with
dimensions: L = 6 in and W = 2 in.?
6 in
A = LW
A = (2 in)(6 in)
2 in
A = 12 in2 or
A = 12 sq in
A = LW works for fractional dimensions as well. The rectangle
in the sketch has a width of 2 ft. and a length of 2.5 ft.
A = LW = (2.5 ft.)(2 ft.) = 5 ft2. If you put the two half squares
at the bottom together, you can see how the area would be 5 ft2.
Notice how the units behave algebraically. In other words,
(ft)(ft) = ft2 just like (x)(x) = x2.
2 ft.
2.5 ft.
A few examples: work out the answer yourself before you click to
the solution.
Example 1. Find the area of a rectangle with a length of 8 ft. and a
width of 3 ft.
Solution: A = LW = (8 ft.)(3 ft.) = 24 ft2.
Example 2. Find the area of a square 3 ft. on each side.
Solution: A = LW = (3 ft.)(3 ft.) = 9 ft2. Since 1 yard is
the same as 3 feet does this mean that 1 yd2. = 9 ft2?
1 yd.
1 yd.
Yes!
A parallelogram is a four-sided figure whose opposite sides are
equal and parallel. Consider the parallelogram in the sketch. The
sides on the top and the bottom are called the base sides of the
parallelogram and the height is the shortest distance between the
two base sides. The height is drawn with a dotted line since it is
not one of the sides of the parallelogram. The height will always
be perpendicular to the two base sides.
h
b
In the next few
slides we are
going to discover
the formula for the
area of a
parallelogram.
Note how the height on the left side of the figure creates a
triangle. If we move this triangle from the left side of the figure
to the right, we will create a rectangle with the same area as the
parallelogram.
b
b
Note how the height on the left side of the figure creates a
triangle. If we move this triangle from the left side of the figure
to the right, we will create a rectangle with the same area as the
parallelogram.
b
b
h
b
Since the area of the new rectangle is A = bh, then the area for
the parallelogram is A = bh.
So whether your parallelogram is tall or short, start with the base.
Then drop a height from the top base to the bottom.
h
h
b
The area of your parallelogram will be A = bh.
b
Now suppose that you were to cut a parallelogram in half by
drawing a diagonal.
Since the area of the parallelogram is defined by A = bh, what
do you think would be a good formula for the area of the
triangle that occupies the lower left half of the parallelogram?
If you answered A = ½ bh give yourself a pat on the back.
Since each triangle is ½ the original parallelogram, it stands
to reason that the area of the triangle would be ½ the area of
the parallelogram.
h
b
Do you think that A = ½ bh would be a good formula for any
triangle? This one for example…
Does it seem reasonable to you that
this triangle is half of some
parallelogram?
If we flip this triangle
and join the two triangles, we get a
parallelogram whose area is defined
by A = bh.
h
The area of the triangle then will be
A = ½ bh.
b
Do you think you can envision every
triangle as half of a parallelogram with
the same height and base as the
triangle?
Before you click, imagine each triangle as half of a parallelogram
with the same height and base.
h
h
b
b
h
h
b
b
h
b
h
b
A trapezoid is a four-sided figure with one pair of sides parallel as
seen below. The parallel sides are called the bases and the shortest
distance from one base of the trapezoid to the other is the height.
The height will always be perpendicular to the bases. In the
sketch, the longer base is denoted B while the shorter is denoted b.
B
h
b
To discover a formula for the area of a trapezoid, watch what
happens when you flip the trapezoid…
and fit the two trapezoids together.
B
b
hh
b
B
To discover a formula for the area of a trapezoid, watch what
happens when you flip the trapezoid…
and fit the two trapezoids together.
You are now looking at a parallelogram whose base is B+b.
B
b
hh
b
B
The area of the parallelogram is A = (B+b)h. Since it took two
trapezoids (with the same area) to make the parallelogram the area
of one trapezoid is A = ½ (B+b)h
A couple of examples. Work them out before you click through.
1. Find the area of a trapezoid with a height of 14 inches, a base of
24 inches and a second base of 30 inches.
24 in
14 in
A = ½ (B + b)h
A = ½ (30in + 24in)14in
A = ½ (54in)14in = 378 in2
30 in
2. Find the area of a trapezoid with a height of 8 meters, one base
of 12 meters and the other of 20 meters.
20m
A = ½ (B + b)h
A = ½ (20m + 12m)8m
8m
A = ½ (32m)8m = 128 m2
12m
We now move our attention to the area of a circle. Thousands of
years ago, mathematicians in Egypt and Greece discovered
something interesting about a circle—regardless of the size, if you
divide the circumference of a circle by its diameter, you always got
the same number. Ever since this irrational number has been
designated by the Greek letter  .
Throughout history this number has mystified mathematicians. 
is an irrational number which means that it cannot be expressed
as the fraction of two integers. 22/7 though is a very good
approximation. The most frequent approximation is 3.14. If you
hit the button on your calculator you will probably see
something like this: 3.141592654. We will be expecting you to
use your calculator for problems that call for  .
We also use  to calculate the area of circle. Do you know this
formula by heart?
A  r
2
r
This is the formula that we will use to
calculate the area of a circle whose
radius is r. We will access  from the
calculator and then round off.
A  r
2
If the radius of the circle above is 5 cm.
then the area will be:
A   5 cm
2
A  25 cm
2
A  25 cm  78.53981cm  78.53cm
2
2
2
Now see what you can do with the following examples.
Example 1. Find the area of the circle with a radius of 12 inches.
Round your answer to the nearest hundredth.
A   r   12 in    144 in 2  452.3893421in 2
2
2
A  452.38 in
2
Example 2. Find the area of the circle with a radius of 1 ft.
Round your answer to the nearest hundredth.
A   r   1 ft   3.141592654ft 
2
2
A  3.14 ft
2
2
What do the circles in these two examples have in common?
Same circle—different units.
Example 3. Find the area of the square with semicircle attached.
See if you can do this on your own before you click again.
Solution. Since the rectangle is a square, all four sides equal 8m.
The top side is the diameter of the semicircle so the radius is 4m.
1
A  LW   r 2
2
1
A  (8m)(8m)   (4m) 2
2
A  64m 2  8 m 2
A  64m 2  25.13274123m 2  89.13m 2
8m
Practice Problems for Area. Find the area of the
indicated figures. (Answers follow.) Round to the
nearest hundredth.
1. Rectangle: L = 152 ft., W = 85 ft.
2. Square with side = 7.24 mm.
3. Parallelogram with base of 5.3 cm and height of 4.6 cm.
4. Rhombus with side = 16.5 in and height = 6.4 in.
5. Triangle with base = 42 cm and height = 16 cm.
6. Right Triangle with legs of 18.3 ft and 28.8 ft.
7. Equilateral triangle with side = 10 ft and height = 8.66 ft.
8. Equilateral triangle with side = 20 ft.
9. Circle with radius of 0.478 ft.
Practice Problems for Area Continued.
10. Circle with diameter = 203 mm.
11. Circle with circumference 12.28 m.
12. Semicircle with diameter = 10 in.
13. Quartercircle with radius = 6 in.
14. A square with attached isosceles triangle in the first figure.
15. A rectangle with attached semicircle in the second figure.
5 in
3 in
7 cm
8 in
24 cm
Practice Problems for Area Continued.
16. The rectangle with quarter-circle removed in the first figure.
17. The square with circle removed in the second figure.
16 m
4m
12 m
4m
18 in
Answers to Practice Problems for Area.
1. A = LW = (152 ft)(85 ft) = 12,920 sq ft
2. A = s2 =(7.24 mm)2 = 52.4176 mm2 ~ 52.42 mm2
3. A = bh = (5.3 cm)(4.6 cm) = 24.38 cm2
4. A = bh = (16.5 in)(6.40 in) = 105.6 in2
5. A = ½ bh = ½ (42 cm)(16 cm) = 336 cm2
6. A = ½ bh = ½ (18.3 ft)(28.8 ft) = 263.52 ft2
7. A = ½ bh = ½ (10 ft)(8.66 ft) = 43.3 ft2
8. A = ½ bh = ½ (20 ft)(17.32 ft) = 173.2 ft2
9.
A  r 2   (.478)2  .7178...  .72 ft 2
More Answers to Practice Problems for Area.
10. A   (101.5mm)2  32365.4729
2 mm2  32365.47 m m2
11. C  12.28 C  2 r  2 r  12.28
12.28
 1.954422701m m
2
2
A   r 2   1.954422701m m  12.00015539m m2
2 r  12.28  r 
A  12.00 m m2
1 2 1
1
2
12. A   r   5 in    25in 2  39.27 in 2
2
2
2
1 2
2
13. A   r  .25 6 in   9 in 2  28.27 in 2
4
1
1
2
14. A  LW  bh  8 in   8 in 3 in   76in 2
2
2
1 2
2
15. A  LW   r  7 cm24 cm  .5 3.5 cm
2
A  187.24 cm2


More Answers to Practice Problems for Area.
1 2
16. A  LW –  r
4
2
 (16 m)(4m) – ¼  4 m   64 m 2 - 12.57m 2
A  51.43m 2
17. A  LW –  r 2  (18in)2 –  (9 in)2
 324in 2 – 254.47in 2
 69.53in 2
By Dr. Julia Arnold
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What is volume?
Volume is a way of measuring space. For example, how much
space is in a rectangular room that has floor measurements of
12 ft. by 16 ft. and a wall or height measurement of 12 ft.
To measure space we use a cube
1 ft. by 1 ft. by 1 ft. or
1 cubic foot.
So, how many of these cubes will
it take to fill the above room?
1 ft
1 ft
1 ft.
Click
for
sound
Click here for floor plan
12
12
16
We can stack 16 times 12 cubes on the floor or 192 cubes and then
we can stack these 192 cubes 12 layers high for a total of 2304 cubes
measured in feet, so we call it cubic feet.
The volume of our room is 12 * 12* 16= 2304 cubic feet
Click
for
sound
A rectangular solid is what you might think of as a box
shape. All the sides are perpendicular to each other and
the three dimensions that it has (length, width, and
height) may be different measurements. Formula for
volume is V = lwh
h
w
l
Click
for
sound
A cube is a rectangular solid in which all of the sides are
equal in length. Formula for volume is V = e3 where e is
the measure of a side.
e
e
e
e
Click
for
sound
A sphere is what you would think of as a ball, no sharp
edges, round all over. Formula for the volume of a
sphere is V  4 r 3
3
where r is the radius of the sphere.
Click
for
sound
A cylinder is what we might think of as a can. While we
may have in mathematics slanted cans, the ones in the
store are what we call a right circular cylinder in that
the sides are perpendicular to the horizontal. The base
and top of the can is a circle and thus has a radius r,
the distance between the top and bottom is called the
height of the can or h. If cut and straightened out this
shape would be a rectangle.
2
V  r h
h
r
Click
for
sound
A right circular cone is similar to an ice cream cone.
In mathematics there are slanted cones, but for our
purposes we will be looking at the right circular cone,
whose base (which is a circle) is perpendicular to the
horizontal.
R is the
radius at
the base of
the cone.
H is the
height of
the cone.
r
h
The formula for
the volume is
1 2
V  r h
3
Click
for
sound
Use the formulas to compute the volume of the
objects in the following problems. When necessary
round your answers to the nearest hundredth. When
writing your final answer, use the appropriate units,
i.e. cu ft. A new convention for writing cubic units or
square units is to use an exponent on the type of unit,
for example; cubic feet would be written ft3.
When finished check your answers.
1. Rectangular solid: L = 73mm, W = 17.2 mm, H = 16 mm
(mm is millimeters)
2. Cube: e = 17.3 in (inches)
In = inches, cm = centimeters, m = meters
3. Sphere: r = 8.2 in
4. Sphere: diameter= 76.4 cm
5. Cylinder: r = 13.5 in, h = 8.2 in
6. Cylinder: d = 16.2 m, h = 7.5 m
7. Cone: r = 1.4 cm, h = 5 cm
8. Cone: d = 9.5 in, h = 7 in
Work out these problems before going to the next slide.
1. Rectangular solid: L = 73mm, W = 17.2 mm, H = 16 mm
(mm is millimeters) 20,089.6 mm3
2. Cube: e = 17.3 in (inches)
3. Sphere: r = 8.2 in
5177.72 in3
2309.56 in3
4. Sphere: diameter= 76.4 cm
233,495.60 cm3
5. Cylinder: r = 13.5 in, h = 8.2 in
6. Cylinder: d = 16.2 m, h = 7.5 m
4694.95 in3
1545.90 m3
7. Cone: r = 1.4 cm, h = 5 cm
10.26 cm3
8. Cone: d = 9.5 in, h = 7 in
165.39 in3
Congratulations! You have just completed
the geometry unit.