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Math Lecture Notes
Geometry Concepts

Find the area of this shape. (§9.2 # 47, Page 536)

area

Perimeter link to here link to index

Three Definitions

There are two common ways to measure the size of a shape: perimeter and area.

Definition

The perimeter of a shape is the distance around its edges.

Definition

The area of a shape is how much room the shape's surface takes up.

We'll start by talking about perimeter.

Let's explore the perimeter of polygons.

Definition

Polygons are closed shapes whose edges are straight lines.

Here are a bunch of polygons: squares, rectangles, triangles, parallelograms, and trapezoids.

polygons

Example 1

Draw a polygon with seven edges.

Example 2

Draw a shape with four edges that is not a polygon.

Adding Up Side Lengths

Perimeter is the distance around a shape. There is nothing more complicated to say, since all polygon perimeter problems are simply adding up the lengths of sides.

perimeter

Finding a perimeter is easy. Just mark a corner (so you don't carelessly forget an edge) and begin adding.

Example 3

Find the perimeter of these two polygons.

polygons

The only possible way to be tricky is demonstrated with this problem:

Example 4

A rectangle has sides 2.5 feet long and 20 inches wide. What is its perimeter in inches?

The trickiness comes from hiding a measurement unit conversion problem inside the perimeter problem.

Area Formulas link to here link to index

Take three paper rectangles to use while we figure out how different area formulas relate.

Area for Rectangles and Squares

Finding the area of a rectangle is easy. The formula Area = length × width is well known to most students.

rectangle area

Squares are just the same.

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Chapter 9 Test, Problem 1: A rectangle has length 9.4 cm and width 7.01 cm. Find its perimeter and area.

Chapter 9 Test, Problem 2: A square has sides of length 4 7/8 inches. Find its perimeter and area.

Chapter 9 Test, Problem 36: A rectangle has length 8 feet and width 3 inches. Find its area in square feet.

Area for Triangles

With a partner, take your paper rectangles and invent a way to use one of them to explain how to measure the area this triangle. Imagine that all you know is how to find the area of a rectangle.

Example 5

How can you make this triangle out of a rectangle? How much of the rectangle does the triangle use?

triangle area

One fold will make this kind of triangle out of a rectangle.

triangle area

We can see the triangle uses up half of the rectangle. So its area is half of the rectangle's area.

The previous triangle had one side straight up and down. How about this other triangle with two slanted sides?

Example 6

How can you make this triangle out of a rectangle? How much of the rectangle does the triangle use?

triangle area

Two folds will make this kind of triangle out of a rectangle.

triangle area

Which still is half of the rectangle, as we can see by moving the small piece over.

triangle area

So for any triangles—whether they have a vertical side or not—we can see the triangle uses up half of the rectangle. Any triangle's area is half of the area of the rectangle that snugly contains it.

For historical reasons we normally do not write Area = 1/2 × length × width for the area of a triangle. Instead of "length" and "width" we call those measuremets "base" and "height".

Why does Area = 1/2 × base × height make people happier? No idea.

Notice that finding the area of a triangle requires its height, not diagonal edge lengths.

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Chapter 9 Test, Problem 4: A triangle has base 8 meters and height 3 meters. Find its area.

Chapter 9 Test, Problem 37: A triangle has base 5 yards and height 3 inches. Find its area in square feet.

Area for Parallelograms

Example 7

How can you make this parallelogram out of a rectangle? How much of the rectangle does the parallelogram use?

parallelogram area

parallelogram area

One cut and slide will make a rectangle out of a parallelogram.

parallelogram area

We could reverse this to make a paralellogram from a rectangle.

The area of a paralellogram is thus the same as the area of the rectangle.

Notice that finding the area of a parallelogram require its height, not diagonal edge length.

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Chapter 9 Test, Problem 3: A parallelogram has base 10 cm and height 2.5 cm. Find its area.

Area for Trapezoids

Example 8

How can you this trapezoid to a single previously studied shape (one rectangle, triangle, or parallelogram)?

Sure, we could cut it apart into two triangles and a rectangle but that is too much work.

trapezoid area

trapezoid area

The area of a trapezoid is thus half of the area of a bigger parallelogram.

Notice that the formula for the area of a trapezoid can be written using an average.

Notice that finding the area of a trapezoid requires both horizontal edges and its height, but not either diagonal edge length.

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Chapter 9 Test, Problem 5: A trapezoid has a bottom 8 feet long, top 4 feet long, and height of 3 feet. Find its area.

Summary Page

If you want to double-check your understanding before the final exam, you can use this summary page. What is the area of each shape? How could you prove it to a first-grader using scissors and perhaps a crayon?

polygon area

Square Units link to here link to index

How does the concept of area relate to measurement unit conversions?

This topic is from §8.7, which concludes chapter eight to make sure you were paying attention to the rest of that chapter.

(You do not need to concern yourself with its typical purple box of conversion rates on page 507.)

Square Feet and Square inches

For the next question, think silently of an answer but do not say anything out loud. Let your classmates have their own moments of insight too. I'll draw a picture to check your answer.

Example 9

How many square inches are in 1 square foot?

square units

We can cut each side into twelve parts. Each cut is one inch apart.

square inches

Each of the smaller squares is one inch per side, so its area is 1 in2.

square inches

How many of these little square inches do we have? 12 × 12 = 144

Note the obvious truth that the perimeter and area of a shape depend upon our choice of measurement units. The same shape will have a different perimeter if we measure around it in inches or miles. The same shape will have a different area if we measure in square inches or square centimeters.

Square Yards and Square Feet

We can repeat that same process for any measurement unit conversion involving square units.

  1. Label each side separately.
  2. Change each side's unit of measurement separately.
  3. Multiply the new measurements to find the new area.

Example 10

How many square feet are in 1 square yard? (Use the three steps above.)

We could also use Unit Analysis. Each of the yard units needs to be canceled separately.

Example 11

How many square feet are in 1 square yard? (Use Unit Analysis.)

When solving these problems it helps to draw a rectangle. We can make up any side lengths that work.

Example 12

How many square miles are in 40,000 square feet?

Square Feet Versus Squared Feet

What is the difference between these two amounts?

22 ft        2 ft2

Sometimes square inches are written abbreviated as sq. in. or in2. Similar abbreviations are used with other square units. Square feet can be sq. ft. or ft2. Square centimeters can be sq. cm. or cm2.

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Chapter 8 Test, Problem 29: Twelve square feet is how many square inches?

Chapter 8 Test, Problem 30: Three square centimeters is how many square meters?

Pi link to here link to index

Three Definitions

Time to talk about about circles. We start with three definitions.

Definition

The perimeter of a circle is called its circumference.

(Don't ask me why. "Perimeter" was already a perfectly usable word for this.)

Definition

The width of a circle, going through the center, is its diameter.

Definition

Half the diameter is the radius.

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Chapter 9 Test, Problem 6: A circle has a radius of one-eighth of an inch. What is its diameter?

Chapter 9 Test, Problem 7: A circle has a diameter of 18 centimeters. What is its radius?

π Hiding in a Circumference

Consider your arm. How many "arm heights" does it take to go around your arm?

pi arm

pi

How about your head? How many "head widths" does it take to go around your head?

pi head

pi

Our answers about measuring our body parts will vary, because arms and heads are not very circular. If we were measuring circles, the answer to the question "How many circle widths does it take to go around the outside?" would be a number bigger than 3 but less than 4.

This answer is roughly 3.14159 but its decimal digits keep on going forever without any repeating pattern. We call this number π (also written as "pi").

Thus we can look really fancy and professional and write a formula Circumference = π × diameter. (Or C = π × d.) But all this formula really says is "the distance around a circle is a bit more than three times its width".

This animation from Wikipedia shows what is happening nicely:

pi animation

π Hiding in an Area

Consider this small square, of area A = r2.

area circle

If we make four of them, the area is then A = 4 × r2.

area circle

Now we ask a second question about circles: how large a circle can we fit into the big square of area 4 × r2?

area circle

It looks like the circle covers more than 3 of the little squares, but not all 4.

So the area of the circle is between A = 3 × r2 and A = 4 × r2.

Actually, the answer is the same number as the answer to the circumference question: π!. The area of a cricle is A = π × r2. Again, this looks fancy and professional but really only says "a circle covers a bit more than three squares with side length equal to the circle's radius".

Beware! Rounding π can cause incorrect answers. To develop optimal math habits always use the calculator key for π so that you do not round in the middle of the problem.

But the textbook will usually use a rounded version of π (either 3.14 or 22/7). This means your answers will not quite match the back of the book because yours are more accurate.

A final note of warning: some students learn the circumference formula as (Some books write the circumference formula as C = π × 2 × r. This makes it look a lot like the area formula: both have a π, an r, and a 2. Don't get mixed up! The easiest way to avoid a careless mistake is to use the circumference formula with a diameter.)