lesson 3 perimeter and area - welcome to the home page...

Free Pre-Algebra Lesson 3 page 1

© 2010 Cheryl Wilcox

Lesson 3

Perimeter and Area

Enclose a flat space; say with a fence or wall. The length of the fence (how far you walk around the edge) is the perimeter, and the measure of the space inside is the area. Finding a Perimeter We know how to measure length. So to find the perimeter of an enclosed space we just measure the edge all the way around. You can measure each straight side separately and add the measurements, or take a long flexible ruler and wrap it all the way around like a ribbon. Either way, the total length around the edge is the perimeter.

Example: Find the perimeter of each figure. Assume measurements are in centimeters.

Count the centimeters around the edge.

The perimeter is 20 cm.

Add up the measurements of the sides. 4 cm + 4 cm + 2 cm + 1 cm + 2 cm + 5 cm = 20 cm

The perimeter is 20 cm.

Finding an Area We’ve measured length with a ruler. But the practical use of space, filling the field with cows, or corn plants, or houses, requires a different kind of measurement. These things are not infinitely thin – they have length AND width. The measure of the space inside the field should take this into account Area can be communicated with non-standard units. People say, “That ranch runs 2000 head of cattle,” or “The theater seats 250 people,” or “The microchip was smaller than a penny.” The cow, the seated person, and the penny use fixed amounts of space to which we can compare other spaces. In mathematics, area is a measure of the size of the space using square units. The number of squares that fill the space is the area.

The Practical Importance of Area and Perimeter

Same Perimeter, Different Area!

If you count the lengths of fencing between the posts, you can see that each farmer used the same amount of fencing material to enclose his field. (Each field has the same perimeter.) But the two fields hold different numbers of cows. (They have different areas.)

Moral: You cannot find the area of a field just by knowing the perimeter.



Comparing Length Units and Area (Square) Units Differentiating between length and area and using the appropriate units for each can be confusing. The square units for area are built from the units of length, but the two are not interchangeable. Sometimes areas have a fixed width (for example, a bolt of fabric is 44 inches wide) and so only the length is used (“give me a yard of that pink satin”). Many students remember that there are formulas to find areas that use lengths for calculation, and so associate areas with lengths. But it’s important to keep the difference clear in your mind conceptually. To say an area is 3 feet is silly; it’s like saying a person is 3 feet old. A foot is not a unit of time nor is it a unit of area. Standard units of length are:

inches, feet, yards, miles (U.S. system)

millimeters, centimeters, meters, kilometers (metric system).

Standard units for area are:

square inches, square feet, square yards, square miles; but also acres (1 acre is 43,560 square feet)

square millimeters, square centimeters, square meters, square kilometers; also hectares.

Figures on a Grid The easiest way to see and count perimeter and area is when a figure has straight sides and is placed on a grid of square units. (Think of a floor tiled with squares of linoleum.) The area is the number of squares. The lines of the grid can be used as rulers to measure the sides. Example: Find the perimeter and area of the space enclosed by the figure.

The size of the picture is reduced, but the original grid is in square centimeters.

To find the perimeter, find the length around the figure.

Count every length – corners have more than one edge!

The perimeter of this figure is 20 cm.

To find the area, count the squares inside the figure.

The area of this figure is 12 square cm.

Units of Length and Square Units of Area

Centimeters

On the ruler, only the edge matters. We’re measuring length, and disregarding width.

Square Centimeters

To find area, we use square units. Each square has length 1 cm and width 1 cm, and has area 1 square centimeter.



The Perimeter of a Rectangle Because a rectangle has two sets of equal sides, we can do less measuring work to find the perimeter. When one length and one width have been measured, we don’t have to keep measuring, because we know the sides across from each other are equal. Walking around the rectangle (in our mind), beginning at the top left, we’d find the perimeter by adding

Length + Width + Length + Width

On the other hand, we could add the two lengths and then the two widths: 2(Length) + 2(Width)

+

Or we could add the length and width, then double that: 2(Length + Width)

+

All these different approaches should yield the same result: the perimeter of the rectangle.

Example: Find the perimeter of the rectangle.

Just for a reference, let’s count the perimeter as in any other figure.

Now, let’s just count the length and width. (This is only half as much counting as before.)

By counting units of length all the way around,

the perimeter is 14 cm.

Using the length and width,

By counting the units, find the length is 4 cm and the width is 3 cm.

Length + Width + Length + Width 2(Length) + 2(Width) 2(Length + Width) 4 cm + 3 cm. + 4 cm + 3 cm

= 14 cm

2( 4 cm ) + 2( 3 cm ) 8 cm + 6 cm

= 14 cm

2( 4 cm + 3 cm) 2 ( 7 cm)

= 14 cm

Rectangles

In a rectangle, the sides across from each other are equal. The measurement of the top or the bottom side is called the length of the rectangle. The measure-ment of the left or the right side is called the width of the rectangle.



The Area of a Rectangle Because of the special properties of rectangles, we can find the area in an easier way than counting. (Although it’s pretty simple to count small areas on a grid, it’s boring to count very large areas, like 43,560 square feet.) This is the same rectangle with length 4 cm and width 3 cm used in the perimeter example, but with the grid revealed, to make finding the area easier.

To find the area, count the squares.

Or notice 3 rows of 4 squares each.

Or notice 4 columns of 3 squares each.

The area is

12 square cm. 3 ( 4 square cm) = 12 square cm

4 ( 3 square cm) = 12 square cm

The reason there are 4 squares in each row is because the length of the rectangle is 4 cm.

The reason there are 3 rows is because the width of the rectangle is 3 cm.

This is why we can calculate the area of a rectangle by multiplying Length x Width.

Area = Length x Width Example: Find the area of a rectangle with length 6 cm and width 4 cm.

We could draw the rectangle on a grid and count the squares:

There are 4 rows with 6 squares in each row: 24 squares.

The area is 24 square centimeters.

Or we can find the area by multiplying Length x Width:

Length x Width (6 cm) (4 cm)

24 square cm

Notice that there is no need to sketch the rectangle.

Keep Paying Attention to the Units

When we multiply 3 rows of 4 squares, it’s clear that the units are squares.

But if you calculate the area using the shortcut

Length x Width (4 cm) (3 cm)

you’re using length and width, measured in centimeters, to find an area, measured in square centimeters. This is a major reason many people are confused about the difference between length units and area units.

When lengths and widths are added, (as when we found a perimeter), the result is also a length.

4 cm + 3 cm = 7 cm

But when lengths and widths are multiplied, the units become units of area.

(4 cm) (3 cm) = 12 square cm



Formulas The shortcuts we reasoned out and used to find the area and perimeter of a rectangle by measuring only the length and width are called formulas. Instead of telling how to calculate the perimeter and area of a specific rectangle, a formula gives the general calculation for any rectangle. We don’t have to remember our whole thought process every time we want to find an area or perimeter, we can just use the formula. Formulas are a good way to start thinking about algebra. If we abbreviate the words to just their initials, things suddenly look quite mathematical:

Perimeter = 2(Length) + 2(Width) P = 2L + 2W

and

Area = Length x Width A = LW

Notice how condensed the algebraic notation is. The words are now just single letters. Multiplication is shown by putting the number right next to the letter, or the two letters right next to each other. But the algebraic formula still represents the same calculation as the formula given in words. If you have a rectangle, and you know the measurements for length and width, you can find the area and perimeter by “plugging” those measurements into the formula, just as we did in the examples before.

Formula Summary

Perimeter = 2(Length) + 2(Width)

P = 2L + 2W

or

Perimeter = 2(Length + Width)

P = 2(L + W)

Area = Length x Width

A = LW

Using the Formulas

Perimeter = 2(6 in) + 2(5 in) = 12 in + 10 in = 22 in

or

Perimeter = 2(6 in + 5 in) = 2 ( 11 in ) = 22 in

Area = 6 in x 5 in = 30 square inches



Lesson 3: Perimeter and Area

Worksheet Name

1. Find the area and perimeter of the figure. Units are centimeters and square centimeters.

The perimeter of the figure is .

The area of the figure is .

2. Find the area and perimeter of the figure. Units are feet and square feet.

The perimeter of the figure is .

The area of the figure is .

3. Draw a rectangle with length 4 cm and width 2 cm. Find the area and perimeter of the rectangle.

The perimeter of the rectangle is .

The area of the rectangle is .

4. Draw a rectangle with length 2 cm and width 4 cm. Find the area and perimeter of the rectangle.

The perimeter of the rectangle is .

The area of the rectangle is .

5. Explain how the rectangles in #3 and #4 are related to each other.

Does it matter which side is called the length and which side is called the width when you are figuring out the area? How about the perimeter?

Did you remember to write the units with your answers?



6. A formula for the perimeter of a rectangle is given in words. Write the formula in algebraic notation.

Perimeter = 2(Length) + 2(Width)

7. The formula for the area of a rectangle is given in algebraic notation. Translate it into words.

A = LW

8. Use the formula to calculate each perimeter. Be sure to include units in your answers.

a. A rectangle with length 5 cm and width 8 cm.

b. A rectangle with length 45 inches and width 60 inches.

c. A rectangle with length 240 miles and width 125 miles.

9. Use the formula to calculate each area. Be sure to include units in your answers.

a. A rectangle with length 5 cm and width 8 cm.

b. A rectangle with length 45 inches and width 60 inches.

c. A rectangle with length 240 miles and width 125 miles.

10. You’re finally fixing up your yard!

a. You’re putting in a new fence all the way around. You need to know: perimeter or area?

b. You’re adding a deck in that space just outside the door. You need to know: perimeter or area?

c. But the deck needs a railing so no one falls off. You need to know: perimeter or area?

d. The back of the house needs paint. One gallon covers 350 square feet. You need to know: perimeter or area?

11. Discuss: You can convert 4 feet to inches by multiplying: Since there are 12 inches in each foot, 4 feet contain 4 x 12 inches, so 4 feet = 48 inches. Why can’t you convert 4 feet to square feet?




Lesson 3

Challenge Worksheet Name

1. The dog pen shown below consists of eight 2-ft long panels that can be arranged in a variety of configurations.

What is the perimeter of the dog pen?

2. Here are two other possible ways to set up the pen. Find the area of each. (Each panel is 2 feet long.)

Area: Area:

3. The pen in the configuration shown in the advertising picture is a regular octogon, (and yes, there is a formula to find its area when the length of the side is known).

Use the grid to estimate the area enclosed by the octagon. (Try to combine partial squares to make whole squares to count.)

Each square in the grid represents one square foot.

The area of the octagon is about .

9. Fill in the blanks:

If the perimeter is a fixed length, narrow shapes have

larger the same as smaller

area than when the shape is rounded.

Why do you think the manufacturer shows the octagon arrangement?

Suppose the sides didn’t have to be straight lines, and the fence for the pen could be any shape. What shape do you think would enclose the greatest area?


http://www.google.com/products/catalog?num=20&hl=en&safe=off&rlz=1G1GGLQ_ENUS274&q=dog+pen&um=1&ie=UTF-8&cid=17935162401446314487&ei=6-cCTMD_CpPMNerMpDs&sa=X&oi=product_catalog_result&ct=result&resnum=3&ved=0CEUQ8wIwAg

http://en.wikipedia.org/wiki/Octagon




Homework 3A Name

1. Estimate the number of pieces in the photo of the stack of firewood.

2. Use the place value table if you need it:

a. Write the number 20,000 in words.

b. Write out the number “two million” using digits.

c. Underline the digit in the hundred billions place in the number 234,699,023,000.

3. Round

a. 234,699,023,000 to the nearest ten million.

b. 234,699,023,000 to 2 significant figures (sig figs)

c. Using your judgment:

“Last year in the U.S. 936,923 people died of cardiovascular disease.”

4. Measure to the nearest cm and nearest inch.

a. The photo of firewood in #1.

length: cm in

width: cm in

b. The outer, largest rectangle enclosing all the homework problems on this page.

length: cm in

width: cm in

Although the size is reduced to fit the page, assume all the grids in this homework are in square centimeters.

5. Find the perimeter and area of the figure.

Perimeter: Area:


Perimeter: Area:

7. Rewrite the formula for the perimeter of a rectangle in algebraic notation.

Perimeter = 2 ( Length + Width )

8. Write the formula to find the area of a rectangle, either in words or in algebraic notation.



9. Draw a rectangle on the grid that has width 4 cm and length 4 cm. Find the perimeter and area.

Perimeter: Area:

10. Draw a rectangle on the grid that has length 7 cm and width 3 cm. Find the perimeter and area.

Perimeter: Area:

11. Find the perimeter of each rectangle.

a. Length 18 cm., Width 22 cm.

b. Length 25 feet, Width 30 feet

c. Length 300 miles, Width 95 miles

d. Length 45 yards, Width 13 yards.

e. Length 3 million km, Width 4 million km.

12. Find the area of each rectangle.






13. Draw a line that is 5 cm long, then measure to the nearest inch: __________________

Units! Units! Check your units!




Homework 3A Answers

1. Estimate the number of pieces in the photo of the stack of firewood.

about 60 pieces


a. Write the number 20,000 in words.

twenty thousand

b. Write out the number “two million” using digits. 2,000,000

c. Underline the digit in the hundred billions place in the number 234,699,023,000.

3. Round

a. 234,699,023,000 to the nearest ten million.

234,700,000,000

b. 234,699,023,000 to 2 significant figures (sig figs)

230,000,000,000


“Last year in the U.S. 936,923 people died of cardiovascular disease.”

937 thousand; more than 900 thousand; nearly 1 million…


a. The photo of firewood in #1.

length: 7 cm 3 in

width: 4 cm 2 in

b. The outer, largest rectangle enclosing all the homework problems on this page.

length: 17 cm 8 in

width: 20 cm 7 in



Perimeter: 20 cm Area: 18 square cm



7. Rewrite the formula for the perimeter of a rectangle in algebraic notation.

Perimeter = 2 ( Length + Width )

P = 2(L + W)

8. Write the formula to find the area of a rectangle, either in words or in algebraic notation.

Area = Length x Width A = LW






Perimeter: 20 cm Area: 21 square cm:



P = 2(L + W) = 2(18 cm + 22 cm)

= 2(40 cm) = 80 cm.

You could use P = 2L + 2W for any of the perimeter problems if you prefer.


P = 2(L + W) = 2(25 ft + 30 ft)

= 2(55 ft) = 110 ft.


P = 2(L + W) = 2(300 mi + 95 mi)

= 2(395 mi) = 790 mi.


P = 2(L + W) = 2(45 yd + 13 yd)

= 2(58 yd) = 116 yd.


P = 2(L + W) = 2(3,000,000 km + 4,000,000 km) = 2(7,000,000 km)

= 14,000,000 km. = 14 million km.



A = LW

= (18 cm)(22 cm) = 396 square cm.


A = LW

= (25 ft)(30 ft) = 750 square ft.


A = LW

= (300 mi)(95 mi) = 28,500 square mi.


A = LW

= (45 yd)(13 yd) = 585 square yd.


A = LW = (3,000,000 km)(4,000,000 km)

= 12,000,000,000,000 square km. = 12 trillion square km.

13. Draw a line that is 5 cm long, then measure to the nearest inch: about 2 inches





Homework 3B Name

1. Estimate the number of cobblestones in the photo.


a. Write the number 19,000,000 in words.

b. Write out the number “thirteen thousand” using digits.

c. What is the place value of the digit 5 in the number 454,444,444,000?

3. Round

a. 89,876,453,021 to the nearest ten billion.

b. 457,328 square cm. to 2 significant figures (sig figs)


“Disneyworld’s average attendance last year was 44,410 visitors per day.”


a. The box around this problem.

length: cm in

width: cm in

b. The box around problem #11 on the next page.

length: cm in

width: cm in



Perimeter: Area:


Perimeter: Area:

7. Rewrite the formula for the perimeter of a rectangle in words.

P = 2( L + W )

8. Write the formula for the area of a rectangle in both words and algebraic notation.




Perimeter: Area:


Perimeter: Area:


a. Length 14 in., Width 21 in.



d. Length 64 m., Width 72 m.

e. Length 70 thousand miles, Width 3 thousand miles.


a. Length 14 in., Width 21 in.



d. Length 64 m., Width 72 m.

e. Length 70 thousand miles, Width 3 thousand miles.

13. Draw a line that is 7 inches long, then measure to the nearest cm: __________________


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