Naming 2-D Shapes

Naming 2-D Shapes

What is a Polygon?

A closed plane figure made up of several line segments that are joined
together. The sides do not cross each other. Exactly two sides meet at every
vertex.

Types
of Polygons

Regular - all angles are equal and all sides are the same length.
Regular polygons are both equiangular and equilateral.

Equiangular - all angles are equal.

Equilateral - all sides are the same length.

	Convex - a straight line drawn through a convex polygon crosses at most two sides. Every interior angle is less than 180°.
	Concave - you can draw at least one straight line through a concave polygon that crosses more than two sides. At least one interior angle is more than 180°.

Polygon Formulas
(N = # of sides and S = length from
center to a corner)

Area of a regular polygon = (1/2) N sin(360°/N) S²

Sum of the interior angles of a polygon = (N - 2) x 180°

The number of diagonals in a
polygon = 1/2 N(N-3)

The number of triangles (when you draw all the diagonals from one
vertex) in a polygon = (N - 2)

Polygon Parts

Side - one of the line segments that make up the polygon. Vertex - point where two sides meet. Two or more of these points are called vertices. Diagonal - a line connecting two vertices that isn't a side. Interior Angle - Angle formed by two adjacent sides inside the polygon. Exterior Angle - Angle formed by two adjacent sides outside the polygon.

Special Polygons
Special Quadrilaterals
- square, rhombus, parallelogram, rectangle, and the trapezoid.

Special Triangles -
right, equilateral, isosceles, scalene, acute, obtuse.

Polygon
Names

Generally accepted names

Sides	Name
n	N-gon
3	Triangle
4	Quadrilateral
5	Pentagon
6	Hexagon
7	Heptagon
8	Octagon
10	Decagon
12	Dodecagon

Names for other polygons have been
proposed.

Sides	Name
9	Nonagon, Enneagon
11	Undecagon, Hendecagon
13	Tridecagon, Triskaidecagon
14	Tetradecagon, Tetrakaidecagon
15	Pentadecagon, Pentakaidecagon
16	Hexadecagon, Hexakaidecagon
17	Heptadecagon, Heptakaidecagon
18	Octadecagon, Octakaidecagon
19	Enneadecagon, Enneakaidecagon
20	Icosagon
30	Triacontagon
40	Tetracontagon
50	Pentacontagon
60	Hexacontagon
70	Heptacontagon
80	Octacontagon
90	Enneacontagon
100	Hectogon, Hecatontagon
1,000	Chiliagon
10,000	Myriagon

To construct a name, combine the
prefix+suffix

Sides Prefix 20 Icosikai... 30 Triacontakai... 40 Tetracontakai... 50 Pentacontakai... 60 Hexacontakai... 70 Heptacontakai... 80 Octacontakai... 90 Enneacontakai...

+

Sides Suffix +1 ...henagon +2 ...digon +3 ...trigon +4 ...tetragon +5 ...pentagon +6 ...hexagon +7 ...heptagon +8 ...octagon +9 ...enneagon

Examples:

46 sided polygon - Tetracontakaihexagon

28 sided polygon - Icosikaioctagon

However, many people use the form
n-gon, as in 46-gon, or 28-gon instead of these names.

History of Hindu-Arabic Numerals

History of Hindu-Arabic Numbers

Zero to Nine

Our Number System

When we study mathematics, we do not often stop to think about our number system.

Where did it come from? How is it different from other number systems, past and present?

The number system we use was invented by the Hindus, and became known to the rest of the world through the Arabs. You may have read about the prophet Mohammed, the founder of the religion Islam. Mohammed lived in Arabia (now called Saudi Arabia). When he died in A.D. 632, he was succeeded by leaders called caliphs. The caliphs sent Arab horsemen out from Arabia in all directions, to invade lands and spread Islam. At the beginning of the eight century, Arab invaders came as far as north-west India. There they came in contact with Hindus, and learned many things from them. Among the things they learned was Hindu number system. When they went back to Arabia, they told their caliph, Harun-al-Rashid, what they had learnt. He was very interested, and he ordered Hindu mathematical books to be translated into Arabic. Later, the books were translated from Arabic into European languages. The Hindu number system is now widely used all over the world, because it is such a good one. Because the Arabs helped to spread it to the rest of the world, it is called the Hindu-Arabic System.

The Hindus decided on a different symbol for each number from one to nine. They numbers a place value, putting single numbers on the right, 10’s one place to the left, 100’s two places to the left, 1000’s three places to the left, etc. so 1888 could be written in only four numerals. Isn’t this easier than Egyptian and Roman systems?

One of the most important inventions of the Hindus in mathematics was the invention of zero. Zero allows us to leave an empty column when we write a number. If there were no zero, we couldn’t distinguish between 123, 1023, 1203, 1230 or 123,000,000. Zero keeps numbers in their places.

In our number system, we use ten as base. A number in one column has ten times the value of the same number in the next column to the right.

Why do you think we use ten as a base? This dictionary definition of the word ‘digit’ may help to explain:

            Digit: a finger or toe, a figure used in arithmetic to represent a number. (Latin digitus, finger, toe)

If we all had twelve fingers and toes, the base would probably have been twelve. We naturally use our fingers for counting. The people of an American Indian tribes in California count in fours, because they count the spaces between their fingers instead of fingers themselves, but most people count their fingers.

It is very useful to have a base for counting. Using only the figures 0 to 9, and arranging them in different ways, we can write any possible number. Imagine how difficult it would be if there a different word or figure for every number up to a hundred.

This is the word for ninety-nine in the African Basuto language:

machoumearobilengmonoolemongametsoarobilengmonoolemong

Which is easier, that or 99?

A number system which uses ten as a base is called a decimal system.

A system which uses twenty (fingers and toes) as a base is called a vigesimal system.

In Dzongkha, both the decimal and vigesimal systems are used.

Primitive people, who did not have many possessions, or who did not have to make big calculations, did not need a developed number system. Some primitive tribes only used the number one and two. Here, for example, is the method of counting of one of the Papuan languages in the area of the Torres Straits between Papua New Guinea and Australia:

1                    urapun

2                    okosa

3                    okosa urapun

4                    okosa okosa

5                    okosa okosa urapun

6                    okosa okosa okosa

Any number above six is called simply called ‘a lot’.

This system is called a binary system, because it is based on two. A binary system is also used in modern computers.

As we have seen, the reason why ten is the base of our number system is probably because we have ten fingers. We often use our fingers as an aid to counting.

What aids can we use? Again, a dictionary definition of a word gives us a clue:

Calculate: to count; to think mathematically. (Latin calculare, to count with the help of little stones. Calculus, a small stone).

There are millions of numbers. So many to spare,

That if you could count every insect in air,

The moth, the mosquito, the bees and the gnat,

There still would be even more numbers than that!

There’s no end to numbers! But don’t be afraid!

There only are ten out of which they are made.

Learn from zero to nine, and the rest will come pat,

For the numbers of numbers all come out of that!

                                               

                                                          Eleanor Farjeon

Why Do We Reciprocate the Divisor?

Rules for Writing Roman Numerals

Rules for Writing Roman Numerals

Roman Numerals

Try the Roman Numeral Challenge.

Roman numerals are expressed by letters of the alphabet:

I=1

V=5

X=10

L=50

C=100

D=500

M=1000

There are four basic principles for reading and writing Roman numerals:

• 1. A letter repeats its value that many times (XXX = 30, CC = 200, etc.). A letter can only be repeated three times.
• 2. If one or more letters are placed after another letter of greater value, add that amount.

VI = 6 (5 + 1 = 6)

LXX = 70 (50 + 10 + 10 = 70)

MCC = 1200 (1000 + 100 + 100 = 1200)

3. If a letter is placed before another letter of greater value, subtract that amount.

IV = 4 (5 – 1 = 4)

XC = 90 (100 – 10 = 90)

CM = 900 (1000 – 100 = 900)

Several rules apply for subtracting amounts from Roman numerals:

• a. Only subtract powers of ten (I, X, or C, but not V or L)

For 95, do NOT write VC (100 – 5). 
DO write XCV (XC + V or 90 + 5)

• b. Only subtract one number from another.

For 13, do NOT write IIXV (15 – 1 - 1). 
DO write XIII (X + I + I + I or 10 + 3)

• c. Do not subtract a number from one that is more than 10 times greater (that is, you can subtract 1 from 10 [IX] but not 1 from 20—there is no such number as IXX.)

For 99, do NOT write IC (C – I or 100 - 1). 
DO write XCIX (XC + IX or 90 + 9)

• 4. A bar placed on top of a letter or string of letters increases the numeral's value by 1,000 times.

XV = 15,  = 15,000

One	I	Eleven	XI	Thirty	XXX
Two	II	Twelve	XII	Forty	XL
Three	III	Thirteen	XIII	Fifty	L
Four	IV	Fourteen	XIV	Sixty	LX
Five	V	Fifteen	XV	Seventy	LXX
Six	VI	Sixteen	XVI	Eighty	LXXX
Seven	VII	Seventeen	XVII	Ninety	XC
Eight	VIII	Eighteen	XVIII	One hundred	C
Nine	IX	Nineteen	XIX	Five hundred	D
Ten	X	Twenty	XX	One thousand	M

Fact Monster/Information Please® Database, © 2007 Pearson Education, Inc. All rights reserved.

Roman Numerals 1-100
1= I  2 = II  3 = III  4 = IV  5 = V  6 = VI  7 = VII  8 = VIII  9 = IX  10 = X  11 = XI  12 = XII  13 = XIII  14 = XIV  15 = XV  16 = XVI  17 = XVII  18 = XVIII  19 = XIX  20 = XX  21 = XXI  22 = XXII  23 = XXIII  24 = XXIV  25 = XXV	26 = XXVI  27 = XXVII  28 = XXVIII  29 = XXIX  30 = XXX  31 = XXXI  32 = XXXII  33 = XXXIII  34 = XXXIV  35 = XXXV  36 = XXXVI  37 = XXXVII  38 = XXXVIII  39 = XXXIX  40 = XL  41 = XLI  42 = XLII  43 = XLIII  44 = XLIV  45 = XLV  46 = XLVI  47 = XLVII  48 = XLVIII  49 = XLIX  50 = L	51 = LI  52 = LII  53 = LIII  54 = LIV  55 = LV  56 = LVI  57 = LVII  58 = LVIII  59 = LIX  60 = LX  61 = LXI  62 = LXII  63 = LXIII  64 = LXIV  65 = LXV  66 = LXVI  67 = LXVII  68 = LXVIII  69 = LXIX  70 = LXX  71 = LXXI  72 = LXXII  73 = LXXIII  74 = LXXIV  75 = LXXV	76 = LXXVI  77 = LXXVII  78 = LXXVIII  79 = LXXIX  80 = LXXX  81 = LXXXI  82 = LXXXII  83 = LXXXIII  84 = LXXXIV  85 = LXXXV  86 = LXXXVI  87 = LXXXVII  88 = LXXXVIII  89 = LXXXIX  90 = XC  91 = XCI  92 = XCII  93 = XCIII  94 = XCIV  95 = XCV  96 = XCVI  97 = XCVII  98 = XCVIII  99 = XCIX  100 = C