Geometry

Geometry: Part IVArea and Volume

ByDick Gill, Julia Arnold and

Marcia Tharpfor

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.

1 ft.

2 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.

2 ft.

3 ft.

2 ft

4 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.

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?

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.

Are you ready to predict the formula for the area of a rectangle? Think before you click.

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

3 cm

A = LW

A = (5 cm)(3 cm)

A = 15 cm2 or

A = 15 sq cm

2 in

6 inA = LW

A = (2 in)(6 in)

A = 12 in2 or

A = 12 sq in

OK, how about a rectangle with dimensions: L = 6 in and W = 2 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.

Solution: A = LW = (8 ft.)(3 ft.) = 24 ft2.

Example 1. Find the area of a rectangle with a length of 8 ft. and a width of 3 ft.

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!

b

h

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.

In the next few slides we are going to discover the formula for the area of a parallelogram.

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

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

h

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.

b b

Then drop a height from the top base to the bottom.

h

h

The area of your parallelogram will be A = bh.

Now suppose that you were to cut a parallelogram in half by drawing a diagonal.

b

h

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.

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.

b

h The area of the triangle then will be A = ½ bh.

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

b

h

b

h

b

h

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

b

h

To discover a formula for the area of a trapezoid, watch what happens when you flip the trapezoid…

B

b

h

b

B

h

and fit the two trapezoids together.

To discover a formula for the area of a trapezoid, watch what happens when you flip the trapezoid…

B

b

h

b

B

h

and fit the two trapezoids together.

You are now looking at a parallelogram whose base is 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.

14 in24 in

30 in

A = ½ (B + b)h

A = ½ (30in + 24in)14in

A = ½ (54in)14in = 378 in2

2. Find the area of a trapezoid with a height of 8 meters, one base of 12 meters and the other of 20 meters.

8m

12m

20m A = ½ (B + b)h

A = ½ (20m + 12m)8m

A = ½ (32m)8m = 128 m2

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 .

We also use to calculate the area of circle. Do you know this formula by heart?

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 .

2rA 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.

r

If the radius of the circle above is 5 cm. then the area will be:

222

2

2

2

53.7853981.7825

25

5

cmcmcmA

cmA

cmA

rA

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.

2

2222

38.452

3893421.45214412

inA

inininrA

Example 2. Find the area of the circle with a radius of 1 ft. Round your answer to the nearest hundredth.

2

222

14.3

141592654.31

ftA

ftftrA

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.

8m

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.

222

22

2

2

13.8913274123.2564

864

)4(2

1)8)(8(

2

1

mmmA

mmA

mmmA

rLWA

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.

8 in

3 in5 in

7 cm

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.

12 m

4m

4 m

16 m

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

9.222 72....7178.)478(. ftrA

More Answers to Practice Problems for Area.

2

22

22

2222

2222

2

222

222

24.187

5.35.2472

1.15

76382

18

2

1.14

27.289625.4

1.13

27.39252

15

2

1

2

1.12

00.12

00015539.12954422701.1

954422701.12

28.1228.122

28.122228.12.11

47.32365mm 232365.4729 mm) (101.5 A 10.

cmA

cmcmcmrLWA

ininininbhLWA

inininrA

inininrA

mmA

mmmmrA

mmrr

rrCC

mm

More Answers to Practice Problems for Area.

2

22

222

2

222

2

in 69.53

in 254.47– in 324

in) (9– in) (18 r – LW A 17.

m 51.43 A

m 12.57 - m 64 4 ¼– m) m)(4 (16

r 4

1– LW A 16.

m

By Dr. Julia Arnold

1

12

1

3

2

3

2 1

4

1

4

5

3

2

5

3

2 1

6

4

5

3

2

7

1

6

4

5

1

3

6

8

4

2

7

8 Cubes

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 cube1 ft. by 1 ft. by 1 ft. or1 cubic foot. So, how many of these cubes willit take to fill the above room?

1 ft.

1 ft

1 ft

Clickforsound

16

12

12

Clickforsound

We can stack 16 times 12 cubes on the floor or 192 cubes and thenwe can stack these 192 cubes 12 layers high for a total of 2304 cubesmeasured in feet, so we call it cubic feet.The volume of our room is 12 * 12* 16= 2304 cubic feet

Click here for floor plan

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

Clickforsound

l

wh

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.

Clickforsound

ee

e

e

3

3

4rV

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

where r is the radius of the sphere.

Clickforsound

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.

Clickforsound

hrV 2

hr

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 thehorizontal.

r

h

R is the radius at the base of the cone.

H is the height of the cone.

hr31

V 2

The formula forthe volume is

Clickforsound

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)

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

In = inches, cm = centimeters, m = meters

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.

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

1. Rectangular solid: L = 73mm, W = 17.2 mm, H = 16 mm(mm is millimeters)

2. Cube: e = 17.3 in (inches)

20,089.6 mm3

5177.72 in3

2309.56 in3

233,495.60 cm3

4694.95 in3

1545.90 m3

10.26 cm3

165.39 in3

Congratulations! You have just completedthe geometry unit.