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INTRODUCTION TO GEOMETRY

The Elements

• Knowing that you can’t define everything and that you can’t prove everything, Euclid began by stating three undefined terms:

Point (Straight) Line

Plane (Surface)

Actually, Euclid did attempt to define these basic terms . . .

is that which has no part

is a line that lies evenly with the points on itself

is a plane that lies evenly with the straight lines on itself

Basic Terms & Definitions

• A ray starts at a point (called the endpoint) and extends indefinitely in one direction.

• A line segment is part of a line and has two endpoints.

A B AB

BA AB

• An angle is formed by two rays with the same endpoint.

• An angle is measured in degrees. The angle formed by a circle has a measure of 360 degrees.

vertex

side

side

• 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 name of Polygon3 triangle

4 quadrilateral5 pentagon6 hexagon7 heptagon8 octagon9 nonagon10 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

Quadrilateral

trapezoidkite

parallelogram

rhombus

rectangle

square

Mor

e sp

ecifi

c

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.

perimeter

area

Triangleh

b

a c

Perimeter = a + b + c

Area = bh21

The height of a triangle is measured perpendicular to the base.

Rectangle and Square

w

l

s

Perimeter = 2w + 2l Perimeter = 4s

Area = lw Area = s2

Parallelogram

b

a h

Perimeter = 2a + 2b

Area = hb Area of a parallelogram = area of rectangle with width = h and length = b

Trapezoid

c d

a

b

Perimeter = a + b + c + d

Area =

b

a

Parallelogram with base (a + b) and height = h with area = h(a + b) But the trapezoid is half the parallelgram

h(a + b)21

h

Ex: Name the polygon

3

21

45

6

hexagon

1

2

34

5 pentagon

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?

Perimeter = 5s

35 = 5s

s = 7 inches

s

Recall: a regular polygon is one with congruent sides.

Ex: A parallelogram has a based of length 3.4 cm. The height

measures 5.2 cm. What is the area of the parallelogram?

3.4

5.2

Area = (base)(height)

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)

L = 26 ft

Let L = LengthL

312 = (L)(12)

Check: Area = (Length)(width) = (12)(26)

= 312

Circle

• 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 = 2r

• The area of a circle is A = r2.

r

d

Find the Circumference

• The circumference, C, of a circle is the distance around the circle. C = 2r

• C = 2r• C = 2(1.5)• C = 3cm

1.5 cm

Find the Area of the Circle• The area of a circle is A = r2

• d=2r• 8 = 2r• 4 = r

• A = r2

• A = 2

• A = 16sq. in.

8 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

+=

8

15

Rectangle with width = 8 and length = 15

Half a circle with diameter = 8 radius = 4

Perimeter of composite figure = 38 + 4.

Perimeter of partial rectangle = 15 + 8 + 15 = 38

Circumference of half a circle = (1/2)(24) = 4.

Ex: Find the perimeter of the following composite figure

28

60

42

12

? = a

? = b

60

a 42

60 = a + 42 a = 18

28

b

12

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

8

8

Area of rectangle = (8)(3) = 24

3

8

Area of triangle = ½ (8)(3) = 12

Area of figure = area of the triangle + area of the square = 12 + 24 = 36.

3

Ex: Find the area of the figure4

3.5

4

3.5

Area of rectangle = (4)(3.5) = 14

4

Diameter = 4 radius = 2

Area of circle = 22 = 4 Area of half the circle = ½ (4) = 2

The area of the figure = area of rectangle – cut out area

= 14 – 2 square units.

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?

20

302

What are the dimensions of the big rectangle (grass and walkway)?

Width = 2 + 20 + 2 = 24

Length = 2 + 30 + 2 = 34

The small rectangle has area = (20)(30) = 600 m2.

What are the dimensions of the small rectangle (grass)?

Therefore, the big rectangle has area = (24)(34) = 816 m2.

The area of the walkway is the difference between the big and small rectangles:

20 by 30

Area = 816 – 600 = 216 m2.

2

Find the area of the shaded region10

10

10r = 5

Area of each circle = 52 = 25

¼ of the circle cuts into the square.

But we have four ¼

4(¼)(25) cuts into the area of the square.

Area of square = 102 = 100

Therefore, the area of the shaded region = area of square – area cut out by circles = 100 – 25 square units

r = 5

32

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