Geometry 1

© Laurel Clifford Creative Commons BY-SA

“Where there is matter, there is geometry.”—Johannes Kepler (1571-1630)

“Geometry is the Art of measuring well.”—Peter Ramus (1515-1572)

Geometry Geometry is one tool we use to view our world, and much of the daily problem solving we

do has some geometric aspect. The word GEOMETRY means Earth (GEO) Measure

(METRY), a means of measuring our world. Geometry has application in many fields,

including practical fields such as carpentry and construction, as well as artistic endeavors

such as sculpture and painting. The Greek mathematician Euclid is famous for the Elements,

beginning with a few basic assumptions (postulates) and developing from these assumptions

the principles and theory of what is now known as Euclidean geometry.

Geometry offers us many opportunities to use both inductive and deductive reasoning. We

can also look beyond the limitations of Euclidean geometry to other geometries and analysis

of the world around us. Geometry offers us many opportunities to use both inductive and

deductive reasoning.

Terms and Notation Geometry has its own language and symbols. We begin our survey of geometry as Euclid

did, by considering some simple geometric figures: points, lines and planes, then create

more figures using these as building blocks. Our first figures are called undefined terms, as

we develop an intuitive understanding of them without precise mathematical definitions. The

notation we use is critical for efficient communication. Consider how much easier it is to

write the symbols 𝐴𝐵 ⃡ rather than write the words, “the line which passes through the two

points A and B and continues on forever in either direction!”

Term Figure and Notation Description

Point

A location in space, with no dimension (not measureable).

Indicated with an upper case letter

Line

𝐴𝐵 ⃡

A collection of points that continues forever in two

directions, has one dimension, is straight (not

measureable). Indicated with two points and a line “hat”

Plane

A collection of infinite points that goes on forever in two

dimensions, flat surface with no depth/thickness (not

measurable). Indicated with three points.

Notice that uppercase letters are used to indicate points. A line contains an infinite number

of points, but its notation uses only two points. This notation reflects two of Euclid’s five

postulates on which he built his geometric theory:

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.1

Thus two points are enough to describe a line as between any two points there is one and

only one line that passes through them and extends forever. One point, such as A would not

1 Weisstein, Eric W. "Euclid's Postulates." From MathWorld--A Wolfram Web Resource.

http://mathworld.wolfram.com/EuclidsPostulates.html

2

be sufficient, as there are an infinite number of directions where the line could go, and we

would not know which direction is indicated.

Using these figures, we can create definitions of other figures. Be especially aware of those

figures which are “pieces” of lines, as they use the same two point notation, but will have a

different “hat” on the points, indicating what type of figure they are; the “hat” is like a rank

insignia on a uniform as it tells us exactly what we’re talking about.

Term Figure and Notation Description

Segment

𝐴𝐵̅̅ ̅̅

A finite subset (piece) of a line with two endpoints,

has measurable length (distance)

𝐴𝐵 refers to the distance between points A and B

while 𝐴𝐵̅̅ ̅̅ refers to the segment itself. A lower case

letter next to the segment can also refer to length.

Ray

𝐴𝐵

A piece of a line with one endpoint that continues

on forever in one direction (not measurable)

The notation uses two points to indicate direction.

Angle

∠A

∠𝐵𝐴𝐶 𝑜𝑟 ∠1

Two rays with a common endpoint; can be two

segments with a common endpoint, or created by

intersecting lines or line segments

The notation uses three points or a number to

clarify which angle is discussed.

Greek letters such as,,,, are sometimes used

to indicate the measure of the angle.

Be careful when discussing angles that you use notation to indicate clearly the angle you

reference. In the angle figures above, the angle indicated as A may seem unambiguous, but

the figure could illustrate two different angles:

The angle indicated with the blue arc: or the angle indicated by the orange arc:

Drawing the arc on the angle helps clarify which angle is discussed. The orange angle is also

known as a reflex angle.

Other figures are much more ambiguous. If we looked at the figure with

two intersecting lines and referred to A, it would be unclear exactly

which angle we are talking about, as there are multiple angles at point A.

We use three point notation, using points on other side of the angle and

the vertex, the vertex (corner point) of the angle is the center point in the

notation. Thus the angle with the blue arc is BAC or CAB. Using

the arc in combination with a number (1) also indicates the angle

discussed.

A A

B

C

D

E

1

A A

A

B

C

D

E

1

Geometry 3

We can build geometric figures using the set operations intersection () and union ().

Recall that these operations relate to the Boolean operators AND (intersection) and OR

(union). Visually, the intersection is where the figures cross each other or overlap, what they

have in common. The union includes all the pieces of the figures involved.

Consider the figure at the left. We can see

illustrated one of Euclid’s postulates about

intersecting lines: two lines intersect in a single

point. For example, the result of 𝐺𝐶 ⃡ ∩ 𝐻𝐷 ⃡ is the

point G, since the line through G and C and the

line through H and D cross only at the point G.

The point G is the only point on 𝐺𝐶 ⃡ AND 𝐻𝐷 ⃡ .

If we consider 𝐶𝐵 ∪ 𝐶𝐹 the result would be an

angle, BCF because the union would include

all the points on either of the two rays. We have

two rays with a common endpoint, C, which

creates an angle.

Try it now 1:

Use the figure above to identify the results of the following:

a. 𝐺𝐹̅̅ ̅̅ ∪ 𝐹𝐷

b. 𝐺𝐷̅̅ ̅̅ ∩ 𝐻𝐹̅̅ ̅̅

c. 𝐺𝐷̅̅ ̅̅ ∪ 𝐻𝐹̅̅ ̅̅

d. 𝐶𝐺̅̅ ̅̅ ∪ 𝐶𝐹̅̅̅̅ ∪ 𝐺𝐹̅̅ ̅̅

e. 𝐵𝐼 ⃡ ∩ 𝐺𝐶̅̅ ̅̅

Measurement Throughout your life you quantify things by assigning a numerical value to it: your height as

you grow up, the time that passes during the day, or the memory you’ve used up storing

pictures on your cell phone. Ancient records as far back as 3000 BC show the Egyptians

using careful measurements and geometry in the construction of the pyramids.

A line segment is a piece of a line between two endpoints, thus linear measurement

measures distance between two points. We need some sort of tool with a standardized unit to

measure this distance, such as a ruler with centimeter or inch marks, or the scale on a map.

The ruler below2 illustrates the idea that the distance is measured between two endpoints, and

the segment length is 5 units. Even though the segment does not start at the zero mark, we

can see it lies between the 3 unit mark and 8 unit mark, and 8 units – 3 units = 5 units.

2 Image from CK-12 Geometry, license CC-BY-NC-SA

4

Angles and Measurement We measure segments by measuring the distance between the endpoints, but how do we

measure an angle? When we measure an angle, we are not interested in distance, as the

distance between the sides of the angle vary. Instead, we measure the amount of rotation

(turn) between the sides of the angle. A full turn, like a full circle, is defined as 360º. Why

360? Possibly we inherited 360 from the Babylonian calendar, with 12 months of 30 days.3

A half turn creates a straight line, and thus we

call this angle a straight angle:

As it is half turn, a straight angle measures

180º.

A quarter turn creates a right

angle, which measures 90.

The square “box” in the

corner of the angle indicates

that the angle is a right angle.

Other angles can be classified in relation to these two angles. Acute

angles are angles that measure less than 90º:

Obtuse angles are angles that measure more than 90 but less than

180º:

Previously mentioned reflex angles measure more than 180 but less

than 360º.

Can an angle measure more than 360 and what does that mean?

If 360º is a full circle, then an angle larger than 360 has rotated at

least one full circle and beyond. Consider if an angle has rotated

405º, it has rotated 360 and then 45º more. It would end or

terminate at the same place as a 45 angle. Such angles are called

coterminal.

3 Weisstein, Eric W. "Degree." From MathWorld--A Wolfram Web Resource.

http://mathworld.wolfram.com/Degree.html

Geometry 5

Adjacent angles share a vertex and a side (it helps to

remember that “adjacent” means “next to”). The total

measure of an angle created by two adjacent angles is the

sum of the measures of each individual angle. In the figure

at the right, mDAB = mDAC + mCAB (the “m”

indicates the measure of the angles), thus mDAB = 25.59º

+ 25.94 = 51.53º.

Two special cases of angle pairs that interest us are supplementary and complementary

angles. Supplementary angles are two angles whose sum

equals 180. If one angle measures 60º then its supplement

measures 120. In the illustration, we can see if two

supplementary angles are also adjacent angles, they form a

straight angle.

Complementary angles are two angles whose sum equals 90º. If one angle

measures 60º then its complement measures 30. From the illustration, we

can see that if two complementary angles are also adjacent angles, they form

a right angle.

Intersecting lines form adjacent angles and opposite

angles. In the figure at the right, we can see that 1

and 2 are adjacent angles, while 1 and 3 are

opposite each other, called vertically opposing

angles or vertical angles. If you examine the figure

closely, you may notice that these angles’ measures

relate to each other in some interesting ways. We

can draw and measure multiple examples or view a

computer generated example which we can

manipulate at http://www.mathopenref.com/anglesvertical.html and use inductive reasoning

to conclude that m1 = m3, and m2 = m4. In general, we can state that vertical angles

have equal measures.

We can also deductively prove this concept without using specific examples or

measurements. We know that 1 and 2 are adjacent angles and form a straight line. Thus

we know that m1 + m2 = 180. Similarly, 2 and 3 are adjacent angles and form a

straight line. Thus we know that m2 + m3 = 180. Since both angle sums equal 180,

then they must equal each other:

m1 + m2 = m2 + m3

Using a little algebra, if we subtract m2 from both sides of this equation, we have:

m1 = m3

We can use a similar argument to show that m2 = m4.

We can apply these angle relationships to find unknown angles created by intersecting lines.

1

2 4

3

6

Try it now 2:

Determine the missing angles a., b., and c. in the given figure:

We can build on this knowledge to explore the angle

relationships created by two parallel lines, coplanar

lines that do not intersect, and a third line that

intersects both, called a transversal (it “transverses”

both lines).

Examining the figure, we can see the vertical angles

we are familiar with, and conclude that m1 = m3,

and m2 = m4. Similarly, we can see that m5 =

m7, and m6 = m8. But how does m1 relate to

m5?

We can again use inductive reasoning and examples drawn or a computer animation such as

the one at http://www.mathopenref.com/transversal.html to determine the relationship. You

may notice that 1 lies in the same location as 5, above the parallel line, and to the right of

the transversal. If we slid the two parallel lines together, 1 and 5 would match up; they

are examples of corresponding angles. From inductive investigation, we can conjecture that

corresponding angles to parallel lines have equal measures.

Applying this thinking to our illustration, we state that m1 = m5, m4 = m8, m2 =

m6, and m3 = m7. Using the “chain rule” of logic (transitive property) we can say:

m1 = m5 = m3 = m7, and m2 = m6 = m4 = m8. We call 4 and 6 alternate

interior angles (as well as 3 and 5) and can state that alternative interior angles to

parallel lines have equal measures. We call 2 and 8 alternate exterior angles (as well

as 1 and 7) and can state that alternate exterior angles to parallel lines have equal

measures.

Putting all our angle relationship ideas together allows us to solve for the missing angles in

more complicated figures.

Try it now 3:

Determine the missing angles a. – g. in the figure given,

assuming that the lines that look parallel are indeed

parallel.

108

a. b.

c.

1 2

3 4

5 6

7 8

62 a.

b. c.

d. e.

f. g.

Geometry 7

We can also use these angle relationships to prove that the sum of the interior angles in a

triangle is 180º. In the figure below, we need to show that m1 + m2 + m3 = 180º.

Assuming the two lines are parallel, and using the

sides of the triangle as transversals, we can

conclude that m1 = m5 since they are

alternate interior angles.

Similarly, m2 = m4.

Notice that m5 + m3 + m4 = 180º since they form a straight angle. Using substitution,

we can take this equation, m5 + m3 + m4 = 180º, replace 5 with 1 and 4 with 2,

and we have m1 + m3 + m2 = 180º, and so the three angles in the triangle add up 180º.

Knowing that the sum of the angles of any triangle is 180º allows us to problem solve with

triangle and other polygon angle sums.

Try it now 4:

Solve for the measures of x, y, and z in the triangles below.

Polygons In combining intersecting lines we created triangles, the simplest polygon. We can build

other polygons using segments and angles. The word polygon comes from the Greek poly-

meaning many and –gon meaning angles. A polygon has the same number sides as angles.

We name polygons based on the number of sides:

Number of

sides

Name of

polygon

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon

12 Dodecagon

n n-gon

1 2

3 4 5

x

30

40 y y

20

z

z

z

8

We previously proved that the sum of the interior angles of any triangle is 180º. We can use

this angle sum for a triangle to find the total interior angle sum for any polygon by dividing

the polygon into triangles. One way to do this process is to draw all the diagonals of the

polygon from a single vertex:

Number of

Sides

3 4 5 6 7 8 … n

Number of

Triangles

Created

1 2 3 4 5 6 … n – 2

Total

Interior

Angle

Sum

180º 180º(2)

=360º

180º(3)

= 540º

180º(4)

= 720º

180º(5)

900º

180º(6)

=1080º

… 180º(n – 2 )

We can see from the table that the number of triangles created by drawing the diagonals from

one vertex is always two less than the number of sides n. We can find the total interior angle

sum by multiplying the number triangles (n – 2) by 180º, 180º(n – 2).

Extending this idea, if we had a decagon, which is a 10-sided polygon, we know that there

would be 8 triangles created, and the total interior angle sum is 180º(8) = 1440º. If the

decagon happened to be a regular polygon, which is a polygon where all the angles and all

the sides are equal, then we could find the measure of each individual angle by dividing the

total measure 1440º by 10 angles, and the result is 144º per angle.

Try it now 5:

Calculate the total interior angle sum of an icosahedron, which has 20 sides. If the

icosahedron was a regular polygon, what is the measure of each interior angle?

Classifying Triangles We classify triangles and quadrilaterals according to the features they have, such as angles:

Right triangle Acute triangle Obtuse triangle

Geometry 9

And sides (the tick marks indicate equal sides, if any):

(no equal sides) (two equal sides) (three equal sides)

Scalene triangle Isosceles triangle Equilateral triangle

Or by both angles and sides:

Right scalene triangle Acute isosceles triangle Obtuse isosceles triangle

The following pair of triangles are congruent triangles, which means they are the same

shape and size. As a consequence, their angles have the same measures, and their sides have

the same length.

So if mABC = 32, then m A’B’C’ = 32

and if AB = 10 cm, then A’B’ = 10 cm.

The triangles below are not congruent, but are similar triangles, which means they are the

same shape, but different sizes. One of them is an enlargement of the other. You may

notice that the angles are equal, but the sides are not. The side lengths are proportional.

So if mABC = 46, then mA’B’C = 46

and if AB = 10 cm, RT = 8 cm, and RS = 7 cm,

we can find the length of AC using

proportional reasoning:

10 𝑐𝑚

8 𝑐𝑚=

𝑥 𝑐𝑚

7 𝑐𝑚

Solve for x either using cross-multiplication or scaling (10/8 multiplied by 7 cm) and AC =

8.75 cm.

A

B C

B’ C’

A’

A

B

C

T

R S

10

Similar triangles show up in application problems where you may not expect them. If you

are standing next to a lamppost, and your shadow is 3 ft long, while you are 5.5 ft tall, and

the lamppost casts a shadow that is 7.5 ft long, how tall is the lamppost?

If you sketch this situation, and visualize a sunbeam creating the

shadow, you can see the triangles involved:

Using proportional reasoning, we have:

5.5 𝑓𝑡 𝑡𝑎𝑙𝑙

3 𝑓𝑡 𝑠ℎ𝑎𝑑𝑜𝑤=

𝑥 𝑡𝑎𝑙𝑙

7.5 𝑓𝑡 𝑠ℎ𝑎𝑑𝑜𝑤

Solving for x via cross-multiplication or scaling, we have x = 13.75

feet, so the lamppost is 13.75 feet tall.

Try it now 6:

A forest service truck is 6 feet tall and casts a 9 foot shadow. It is parked next to a fire

lookout tower that casts a 240 foot long shadow. How tall is the lookout tower?

Classifying Quadrilaterals We classify quadrilaterals by their angle size and side length characteristics as well as

whether they have any parallel sides. A quadrilateral tree helps illustrate these

interrelationships. As we proceed higher in the tree, the quadrilaterals get more specialized,

and every figure higher on the tree has the same features as the figures below it. For

example, a square is a specialized quadrilateral that is both a rhombus and a rectangle.

Quadrilateral: Polygon with 4

sides

Trapezoid:

Quadrilateral

with one pair of

paralles sides

Isosceles Trapezoid: Trapezoid with

nonparallel sides equal

Parallelogram: Quadrilateral

with two pairs

of parallel sides

Kite:

Quadrilateral with

two pairs of adjacent

equal sides

Rhombus:

Quadrilateral with

all equal sides

Rectangle:

Quadrilateral with

all equal angles

Square:

Quadrilateral with

all equal angles and

all equal sides

Geometry 11

Measuring Polygons We’ve measured side lengths and angle rotation. When working with polygons, we can still

measure their side lengths and their angles. We can also measure other aspects of polygons.

Suppose you are building your dream house, and have designed a room that will be your

office/study as shown below. You can assume that since you are a meticulous builder, all

angles that are supposed to be right angles are actually right angles. You decide to carpet

the room, and need to purchase base board trim as well, so you have two questions to deal

with (besides what color to choose):

1. How much carpet will you need to buy?

2. How much trim?

In order to carpet the room, you need to measure the interior

of the room, the floor area. This room is irregularly shaped,

but if you recall how to find the area of a rectangle, we can

divide the room into rectangles and find the area of each

rectangle then add up the individual areas.

If we use the blue dashed line to separate the room into two rectangles, the lower rectangle

measures 5 ft by 12 ft, while the upper rectangle measures 7 ft by 8 ft (subtracting the 5 ft

from the 12 ft to get the remaining 7 ft). Thus the two areas are: (5 ft)(12 ft) = 60 ft2, and (7

ft)(8 ft) = 56 ft2, and 60ft2 + 56 ft2 = 116 ft2. Thus we need 116 ft2 of carpeting.

The baseboard trim goes around the edge of the room, so we need to find the perimeter by

adding up each of the distances around the outside. Using the dimensions given, and finding

the unknown dimensions from the given dimensions with which they are parallel, the

perimeter is 8 ft + 12 ft + 12 ft + 5 ft + 4 ft + 7 ft = 48 ft. Thus we need 48 ft of carpeting.

Looking back, how are the two questions different? How are the answers different? We are

measuring two very different things: the interior plane area and the exterior linear border.

As a result, the units we use are also different. When we found area, our units were ft2,

square feet, while perimeter units were ft, linear feet.

In geometry, we often use linear measurement (measuring pieces of lines) to find the

perimeter of figures. The perimeter can be found by adding up the distances along the

outside of the figure. With some polygons, we can create formulas for perimeter:

P = 2x + 2y

P = 8x P = 3s

12 ft

12 ft

5 ft

8 ft

x

x

x

Regular Octagon

y

y

s

Equilateral Triangle

Rectangle

12

There is a danger in memorizing a formula without understanding the concept of perimeter:

you apply the wrong formula for the information given.

Circles When looking at polygons, notice what happens to

the shape of the polygon as you increase the number

of sides. The image at the left demonstrates the

pattern when you increase the number of sides in

regular polygons (equal sides and equal angles). We

know from our previous work that the total interior

angle sum gets larger as you increase the number of

sides. We can see from this visual image that the

polygons themselves get rounder and rounder,

approaching the shape of a circle.

A circle can be thought of as a regular polygon with

an infinite number of sides. We can measure the

perimeter and area of circles using some of the same

concepts as polygons.

The perimeter of a circle is called its circumference. The radius of a circle is the distance

from the center of the circle to a point on the circle itself. The diameter of a circle is the

distance from a point on the circle through the center to the opposite side. The diameter is

twice the radius in length.

Consider the circle below, created by the geometry program, Geometer’s Sketchpad:

With this particular program, we can drag the circle and change its size. The radius,

diameter, and circumference will change, but the ratio between the circumference and

diameter always stays the same, about 3.14 which you may recognize as an approximation

for .

For any circle, 𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒

𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟=

𝐶

𝑑= 𝜋

If we solve this equation for C, we have: C = d,

And given that d = 2r, C = (2r) = 2r

Circumference BA

Diameter = 3.14

Diameter = 5.29 cm

Radius BA = 2.64 cm

Circumference BA = 16.61 cm

B

A

Geometry 13

Example 1: Problem Solving with Circles

With the circumference equation we can solve for linear measurements involving circles. If

we know that a circle has radius 7 cm, we can find the circumference around the circle:

C = 2(7 cm) = 14 cm, or approximately 43.982 cm.

If we know a circle has circumference 86.8 cm, we can find its diameter:

86.8 cm = d, so d = 86.8 cm/ or approximately 27.629 cm.

The ratio has been studied for millennia. The Hebrews used 3 as an approximation for .

The Babylonians also used 3, but created more precise estimations of its value. An

approximation for is shown on the Rhind Papyrus (1650 BC) of the Egyptians. The Greek

mathematician Archimedes (287–212 BC) and the Chinese mathematician Zu Chongzhi

(429–501 AD) both calculated approximations for using regular polygons.4 Today,

supercomputers calculate to trillions of digits.5

Area We used linear measurement to calculate the amount of baseboard trim we needed for our

new office. When calculating the amount of carpet we needed, we are talking about area

measurement, enclosed in the interior of polygons, measured in two dimensions (length and

width), and measured in square units.

1 linear unit looks like a piece of a line:

1 square unit looks like a section of a plane:

We need to be careful when relating between linear units and square units. We know

conversion factors for linear units, such as 1 foot = 12 inches, but these do not translate

directly to area units: 1 square foot does NOT equal 12 square inches. How many square

inches are in 1 square foot?

If we take a 1 foot by 1 foot square, and divide each side of

the square into 12 inch units, we have a square that is 12

inches by 12 inches, and 144 square inches fit inside this

area, as shown by the 144 squares visible.

We see this idea by multiplying the conversion factors:

1 𝑓𝑜𝑜𝑡

12 𝑖𝑛𝑐ℎ𝑒𝑠×

1 𝑓𝑜𝑜𝑡

12 𝑖𝑛𝑐ℎ𝑒𝑠=

1 𝑓𝑜𝑜𝑡2

144 𝑖𝑛𝑐ℎ𝑒𝑠2

If carpet is sold by the square yard, and we need 116 ft2 of carpet, we must take care with our

conversion factors to convert square feet into square yards. We need to cancel square feet:

116 𝑓𝑡2

1×

1 𝑦𝑎𝑟𝑑

3 𝑓𝑡×

1 𝑦𝑎𝑟𝑑

3 𝑓𝑡 or

116 𝑓𝑡2

1×

1 𝑦𝑎𝑟𝑑2

9 𝑓𝑡2= 12. 8̅ 𝑦𝑎𝑟𝑑2

4 Pi Day: History of Pi | Exploratorium. (n.d.). Pi Day: History of Pi | Exploratorium. Retrieved June 23, 2014,

from http://www.exploratorium.edu/pi/history_of_pi/ 5 Yes, Trillions! Check out: http://www.numberworld.org/misc_runs/pi-5t/details.html

14

When applying proportional reasoning with area, we must also make sure to be comparing

appropriate units and quantities.

Example 2: Proportional Reasoning with Area

If a 12 inch diameter pizza requires 10 ounces of dough, how much dough is needed for a 16

inch pizza?

To answer this question, we need to consider how the weight of the dough will scale. The

weight will be based on the volume of the dough. However, since both pizzas will be about

the same thickness, the weight will scale with the area of the top of the pizza. We can find

the area of each pizza using the formula for area of a circle, 2A r :

A 12” pizza has radius 6 inches, so the area will be 26 = about 113 square inches.

A 16” pizza has radius 8 inches, so the area will be 28 = about 201 square inches.

Notice that if both pizzas were 1 inch thick, the volumes would be 113 in3 and 201 in3

respectively, which are at the same ratio as the areas. As mentioned earlier, since the

thickness is the same for both pizzas, we can safely ignore it.

We can now set up a proportion to find the weight of the dough for a 16” pizza:

22 in 201

ounces 

in 113

ounces 10 x Multiply both sides by 201

10

201113

x = about 17.8 ounces of dough for a 16” pizza.

It is interesting to note that while the diameter is 16

12 = 1.33 times larger, the dough required,

which scales with area, is 1.332 = 1.78 times larger.

There are many formulas for finding areas of polygons. It’s better to develop conceptual

understanding than to just memorize formulas. We can build many area formulas from the

area of a rectangle. You probably can easily recall the formula for the area of a rectangle as

A = lw. What are “l” and “w” and what kind of measurement do they represent, linear or

area? Notice that this formula uses linear measurements to find area. Why does it “work” to

use linear measurements (the dimensions of the rectangle) to find area?

If we consider that we are counting the number of squares that fit

inside the rectangle to find the area, we can see how the

dimensions can count these squares for us.

The rectangle has rows of 9 squares. Theses rows of squares are

stacked 5 high, so if we multiply the dimensions (9×5) we really

are multiplying: 9 𝑠𝑞𝑢𝑎𝑟𝑒𝑠

𝑟𝑜𝑤×

5 𝑟𝑜𝑤𝑠

1= 45 𝑠𝑞𝑢𝑎𝑟𝑒𝑠 𝑡𝑜𝑡𝑎𝑙

When we multiply the base length by the height, we are counting the number of square units

in the figure!

Geometry 15

Area of a rectangle

Because we will be moving on to other polygons, we consider the area of a rectangle

as:

A = (base)(height),

where the base and height are perpendicular (at a right angle) to each other.

This will be our “master formula” for creating formulas for other polygons.

We can apply the same kind of thinking, and “create rectangles” for other polygons. This

will allow us to create more formulas from the “master formula,” A = (base)(height), we

created for the area of a rectangle.

A parallelogram can be thought of as a sheared or tilted

rectangle. If we “cut and paste” a triangle from one side of

the parallelogram to the other side, we create a rectangle, and

our area formula remains the same: A = (base)(height) as

long as we’re careful with the height at a right angle to our

base:

A = (6 units)(5 units) = 30 units2

A triangle can be thought of as half of a rectangle. If we

copy and paste the triangle, we can create a complete

rectangle. So our area formula, A = (base)(height), will need

to be cut in half:

A = (7 units)(6 units)/2 = 21 units2

So the area formula for a triangle is:

A = ½(base)(height)

A trapezoid can be cut up into triangles or other shapes to

find its area. One method illustrated here is to copy and

paste the trapezoid rotated 180 next to itself to create a

parallelogram, and then apply the parallelogram area

formula. Since we use two trapezoids to create the

parallelogram, we will cut the area in half:

A = (6 units + 4 units)(3 units)/2 = 15 units2

Notice that to create the base length, the two parallel sides

(or bases) of the trapezoids are added together, so the area

formula for a trapezoid is: A = ½ (sum of the bases)(height)

16

What about the area of a circle? We can “cut up” a circle to make it approximate a

parallelogram:

(Nice animation of this at

http://www.wku.edu/~tom.richmond/Pir2.html )

Using our area formula A = (base)(height), and the base

of the parallelogram we have created is r, with height

r,

A = (r)(r) = r2

Example 3: Problem Solving with Circle Area

Suppose we have a circle with radius 3.5 cm. We can calculate its area:

A = r2

A = (3.5 cm)2

A =12.25 cm2, or approximately 38.48 cm2.

If we know the area is 100 cm2 and want to find the diameter, we’ll have to work a little

harder, as our formula only relates the radius to the area.

A = r2

100 cm2 = r2, divide by and then square root to undo the square,

5.64 cm r, but we want the diameter, and knowing d = 2r, d 2(5.64 cm) = 11.28 cm.

One more triangle concept… Consider the triangle drawn on the grid. We can find its area:

A = ½ (5 units)(5 units) = 12.5 units2,

using the concept that a triangle is half of a rectangle.

But what happens when we calculate the perimeter?

Adding up the side lengths is usually a straight-forward

process. In this case, P = 5 units + 5 units + …

Here’s where the problem arises: the units along the

diagonal side (the hypotenuse) of the triangle are not the

same size as the units along the two legs. Just eyeballing it suggests that the diagonal

(oblique) units are a bit longer than the horizontal and vertical units. If we are using the grid

as our units, we need a method to calculate the diagonal side.

Pythagorean Theorem

To find the length of the oblique side, which is the hypotenuse in this right triangle,

use the Pythagorean Theorem, which says if a right triangle has legs of lengths a and

b, and hypotenuse c, then a2 + b2 = c2.

Geometry 17

In the case of the previous triangle, we can find the hypotenuse, and then the perimeter:

a = 5 units, b = 5 units, so a2 + b2 = c2 is:

(5 units)2 + (5 units)2 = c2

25 units2 + 25 units2 = c2

50 units2 = c2 (square root to undo the square)

√50 units = c

7.071 units c

And the perimeter of the triangle is: P = 5 units + 5 units + 7.071 units 17.071 units.

We can look at this result we found for “c” and say it is a little more than 7 units (since the

square root of 49 is 7). We can also simplify the square root by considering that:

√50 = √25√2 = 5√2 units

There’s an interesting pattern that appears here because this triangle is a special case: it is an

isosceles right triangle. Since the legs are equal, we can expect this pattern to appear again.

If a right triangle has equal legs “n” then n2 + n2 = c2, and we have 2n2 = c2.

When you solve for “c” by square rooting, we have:

√2𝑛2 = 𝑛√2 = 𝑐 The hypotenuse of an isosceles triangle will always be the leg length times the square root of

2.

Try it now 7:

Solve for “x” in each triangle below:

The first two triangles are examples of Pythagorean Triples. The third is an isosceles

triangle.

a. b. c.

Example 4: And… back to our dream house…

If you have the room shown below, and you want to put down parquet flooring, which comes

in 1 ft by 1 ft squares, how many squares do you need to buy? How

much baseboard for trim?

The flooring is area. We can view this room as a square with a

triangle cut off the corner: A = (12 ft)(12 ft) – ½(4 ft)(4ft) = 136 ft2.

Since flooring is (1 ft)(1 ft) = 1 ft2 squares, we 136 squares.

Baseboard is perimeter. The oblique edge is the hypotenuse of an

isosceles right triangle, and is 4√2 𝑓𝑡 long, so

P = 12 ft + 12 ft + 8 ft + 8 ft + 4√2 𝑓𝑡 45.66 ft of baseboard.

x

17

8

x

24

25

11

11 x

8 ft