INTRODUCTION TO
GEOMETRY
MSJC ~ San Jacinto Campus
Math Center Workshop Series
Janice Levasseur
Geometry
• The word geometry comes from Greek
words meaning “to measure the Earth”
• Basically, Geometry is the study of shapes
and is one of the oldest branches of
mathematics
The Greeks and Euclid
• Our modern understanding of geometry
began with the Greeks over 2000 years
ago.
• The Greeks felt the need to go beyond
merely knowing certain facts to being able
to prove why they were true.
• Around 350 B.C., Euclid of Alexandria
wrote The Elements, in which he recorded
systematically all that was known about
Geometry at that time.
The Elements
• Knowing that you can’t define everything and
that you can’t prove everything, Euclid began by
stating three undefined terms:
 Point
is that which has no part
 (Straight) Line is a line that lies evenly with the points
on itself
 Plane (Surface) is a plane that lies evenly with the
straight lines on itself
Actually, Euclid did attempt to define these basic
terms . . .
Basic Terms & Definitions
• A ray starts at a point (called the endpoint)
and extends indefinitely in one direction.
A
B
AB
• A line segment is part of a line and has
two endpoints.
A
B
AB
• An angle is formed by two rays with the
same endpoint.
side
vertex
side
• An angle is measured in degrees. The
angle formed by a circle has a measure of
360 degrees.
• A right angle has a measure of 90
degrees.
• A straight angle has a measure of 180
degrees.
• A simple closed curve is a curve that we
can trace without going over any point
more than once while beginning and
ending at the same point.
• A polygon is a simple closed curve
composed of at least three line segments,
called sides. The point at which two sides
meet is called a vertex.
• A regular polygon is a polygon with sides
of equal length.
Polygons
# of sides
3
4
5
6
7
8
9
10
name of Polygon
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
Quadrilaterals
• Recall: a quadrilateral is a 4-sided polygon. We can
further classify quadrilaterals:
 A trapezoid is a quadrilateral with at least one pair of
parallel sides.
 A parallelogram is a quadrilateral in which both pairs of
opposite sides are parallel.
 A kite is a quadrilateral in which two pairs of adjacent
sides are congruent.
 A rhombus is a quadrilateral in which all sides are
congruent.
 A rectangle is a quadrilateral in which all angles are
congruent (90 degrees)
 A square is a quadrilateral in which all four sides are
congruent and all four angles are congruent.
From General to Specific
More specific
Quadrilateral
trapezoid
kite
parallelogram
rhombus
rectangle
square
Perimeter and Area
• The perimeter of a plane geometric figure
is a measure of the distance around the
figure.
• The area of a plane geometric figure is the
amount of surface in a region.
area
perimeter
c
a
Triangle
h
b
Perimeter = a + b + c
1
Area = bh
2
The height of a triangle is
measured perpendicular to the
base.
Rectangle and Square
s
w
l
Perimeter = 2w + 2l
Perimeter = 4s
Area = lw
Area = s2
Parallelogram
a
h
b
Perimeter = 2a + 2b
Area = hb
 Area of a parallelogram
= area of rectangle with
width = h and length = b
Trapezoid
a
c
b
d
h
b
a
Perimeter = a + b + c + d
Area =
1
h(a + b)
2
 Parallelogram with base (a + b) and height = h
with area = h(a + b)
 But the trapezoid is half the parallelgram 
Ex: Name the polygon
2
1
6
3
 hexagon
5
4
1
2
5
 pentagon
3
4
Ex: What is the perimeter of a
triangle with sides of lengths 1.5 cm,
3.4 cm, and 2.7 cm?
1.5
2.7
3.4
Perimeter = a + b + c
= 1.5 + 2.7 + 3.4
= 7.6
Ex: The perimeter of a regular
pentagon is 35 inches. What is the
length of each side?
s
Recall: a regular polygon is
one with congruent sides.
Perimeter = 5s
35 = 5s
s = 7 inches
Ex: A parallelogram has a based
of length 3.4 cm. The height
measures 5.2 cm. What is the area
of the parallelogram?
Area = (base)(height)
5.2
3.4
Area = (3.4)(5.2)
= 17.86 cm2
Ex: The width of a rectangle is
12 ft. If the area is 312 ft2, what
is the length of the rectangle?
12
312
Area = (Length)(width)
Let L = Length
L
312 = (L)(12)
L = 26 ft
Check: Area = (Length)(width) = (12)(26)
= 312
r
Circle
d
• A circle is a plane figure in which all points are
equidistance from the center.
• The radius, r, is a line segment from the center of
the circle to any point on the circle.
• The diameter, d, is the line segment across the
circle through the center. d = 2r
• The circumference, C, of a circle is the distance
around the circle. C = 2pr
• The area of a circle is A = pr2.
Find the Circumference
1.5 cm
• The circumference, C,
of a circle is the distance
around the circle. C = 2pr
• C = 2pr
• C = 2p(1.5)
• C = 3p cm
Find the Area of the Circle
• The area of a circle is A = pr2
8 in
• d=2r
• 8 = 2r
• 4=r
• A = pr2
• A = p(4)2
• A = 16p sq. in.
Composite Geometric Figures
• Composite Geometric Figures are made
from two or more geometric figures.
• Ex:
+
• Ex: Composite Figure
-
Ex: Find the perimeter of the
following composite figure
15
8
=
+
Rectangle with width = 8
and length = 15
Perimeter of partial rectangle
= 15 + 8 + 15 = 38
Half a circle with diameter = 8
 radius = 4
Circumference of half a circle
= (1/2)(2p4) = 4p.
Perimeter of composite figure = 38 + 4p.
Ex: Find the perimeter of the
following composite figure
60
12
28
42
?=b
12
28
?=a
60
b
a
42
60 = a + 42  a = 18
28 = b + 12  b = 16
Perimeter = 28 + 60 + 12 + 42 + b + a
= 28 + 60 + 12 + 42 + 16 + 18 = 176
Ex: Find the area of the figure
3
3
3
8
Area of triangle = ½ (8)(3) = 12
8
3
8
Area of rectangle = (8)(3) = 24
Area of figure
= area of the triangle + area of
the square = 12 + 24 = 36.
Ex: Find the area of the figure
4
4
3.5
3.5
Area of rectangle = (4)(3.5) = 14
4
The area of the figure
= area of rectangle – cut out area
= 14 – 2p square units.
Diameter = 4  radius = 2
Area of circle = p22 = 4p  Area of half the circle = ½ (4p) = 2p
Ex: A walkway 2 m wide surrounds
a rectangular plot of grass. The
plot is 30 m long and 20 m wide.
What is the area of the walkway?
2
30
20
What are the dimensions of the big
rectangle (grass and walkway)?
Width = 2 + 20 + 2 = 24
Length = 2 + 30 + 2 = 34
2
Therefore, the big rectangle has area
= (24)(34) = 816 m2.
What are the dimensions of the small rectangle (grass)? 20 by 30
The small rectangle has area = (20)(30) = 600 m2.
The area of the walkway is the difference between the big and small
rectangles:
Area = 816 – 600 = 216 m2.
Find the area of the shaded region
10
10
Area of square =
102 = 100
Area of each
circle = p52 = 25p
r=5
10
r=5
¼ of the circle cuts
into the square.
But we have four ¼
4(¼)(25p ) cuts into
the area of the square.
Therefore, the area of the shaded region
= area of square – area cut out by circles = 100 – 25p square units