Description

a research about sets

Effat University

Project

FALL 2021

ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT

INSTRUCTOR NAME: Dr. Nema Salem

COURSE NAME: Mathematics for Arch. & Design

COURSE #: GMTH141A

SECTION: 01 and 04

Student Name: ——————————-Student No. : ——————————-Date & Time:

InstructorÃ¢â‚¬â„¢s Name:

Dr. Nema Salem

InstructorÃ¢â‚¬â„¢s Signature:

GMTH141A

Instructor Name: Dr. Nema Salem

Page 1 of 2

Effat University

Project

FALL 2021

Question (1)

(5 Points)

1. Students are working in team of three.

2. Select an idea/problem

3. Do literature review about it

4. Describe your methodology in solving it

5. Show your results: analytically, graphically, etc.

6. Write a discussion and then conclude your work

Required

1. Submit a report

2. Present, orally, your work

GMTH141A

Instructor Name: Dr. Nema Salem

Page 2 of 2

Set Theory

CHAPTER 2

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Sets and Elements Notation

Ã¢â€“ÂªGrouping objects together and viewing them as one entity.

Ã¢â€“ÂªSets are noted by Upper Case Letters, like: P, M, X, R

Ã¢â€“ÂªElements are noted by Lower Case Letters, like: p, m, x, r

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Sets and Elements

Ã¢â€“ÂªSet:

Ã¢â€“Âª Any well defined collection of objects,

Ã¢â€“Âª Objects are called the elements of the set.

Ã¢â€“ÂªExamples:

Ã¢â€“Âª Vowels of the Alphabets: P = {a, e, i, o, u}

Ã¢â€“Âª Colors of the Italian Flag: Y = {red, white, green}

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Sets and Elements

Ã¢â€“ÂªPrime numbers:

A = {2, 3, 5, 7, 11, 13, Ã¢â‚¬Â¦Ã¢â‚¬Â¦.} infinite

Ã¢â€“ÂªLetters in the word Ã¢â‚¬Å“hobbyÃ¢â‚¬Â:

R = {h, o, b, y}

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People in a class: { Alice, Maryam, Sophie, Sammy}

Classes offered by a department: {CS 101, CS 202, Ã¢â‚¬Â¦ }

Colors of a rainbow: { red, orange, yellow, green, blue, indigo, purple }

States of matter { solid, liquid, gas}

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Sets and Elements

We symbolize an element belonging to a set as follows:

Example: A= {1, 7 , 5 , 10 , 14},

Element 10 belongs to set A, it can be expressed as:

10 Ãâ€ž A

If an element does not belong to a set, we use the symbol Ã¢Ë†â€°

13 Ã¢Ë†â€° A

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Sets and Elements: Exercises

1. A = {1, 3, 5}

{1} Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦. A

{2} Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦..A

2. B = { } = Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦..

3. C is the set of even numbers greater than 2 and less than 12.

C = {Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦..}

{4} Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦C

{6 }Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦ C

{8 }Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦C

{10} Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦..C

{2} Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦.C

{12} Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦C

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Infinite Sets

Ã¢â€“Âª Infinite Set as odd numbers:

P = {1 , 3 , 5 , 7 , 9 , 11 , 13 , 15 , 17 , Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦..}

Ã¢â€“Âª Set-builder notation:

P = {x| x is an odd integer}

read as;

Ã¢â‚¬Å“P is the set of x such that x is an odd integerÃ¢â‚¬Â

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Infinite Sets: Example

Ã¢â‚¬Â¢ Natural numbers N = {1, 2, 3, Ã¢â‚¬Â¦}

Ã¢â‚¬Â¢ Whole numbers = {0, 1, 2, 3, Ã¢â‚¬Â¦}

Ã¢â‚¬Â¢ Integers Z = {Ã¢â‚¬Â¦, -2, -1, 0, 1, 2, Ã¢â‚¬Â¦}

Ã¢â‚¬Â¢ Positive Integers Z+ = {1, 2, 3, 4, Ã¢â‚¬Â¦}

Ã¢â‚¬Â¢ Rational Numbers = Q = {a/b | aÃ¯Æ’Å½Z , bÃ¯Æ’Å½Z, and bÃ¯â€šÂ¹0}

= {1.5, 2.6, -3.8, 15, Ã¢â‚¬Â¦}

Ã¢â‚¬Â¢ Real Numbers = {47.3, -12, Ã¯ÂÂ°, Ã¢â‚¬Â¦} (rational and irrational numbers)

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Universal Set (U)

Ã¢â€“Âª A large set including all the elements, it depends on the objects we

are studying.

Example:

Ã¢â€“ÂªIf we are studying non-zero positive integers up to 15:

Ã¢â€“Âª The Universal set is {1, 2, 3, 4, 5, 6, 7, 8, 9,10, 11, 12, 13, 14, 15}.

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U = Universal Set

U = {0,1, 2, 3, 4, 5, 6, 7, 8, 9}

U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

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Empty Set

Ã¢â€“Âª It is a set that includes no elements at all, it has a symbol ÃƒËœ

called Ã¢â‚¬ËœPhiÃ¢â‚¬â„¢ as:

Ã¢â€“ÂªThe people living on Mars.

Ã¢â€“ÂªThe people who visited Neptune.

Ã¢â€“ÂªThe number of astronauts who walked in space without a space suit.

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Power Set

The Power Set of set A, denoted is the set of all subsets.

P({1,2, 3}) = {

Ã¯Æ’â€ ,

{1},{2},{3},

{1,2},{1, 3},{2, 3},

{1,2, 3} }

If a set has n elements, then the power set will have 2n elements

Find the Power set of set B = {1,4,8,5}

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Set Builder Notation

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Subsets and Proper Subsets

Ã¢â€“Âª Set A is a subset of set B if every element in set A is also an element in

set B. (A is contained in B) Ã°ÂÂÂ´ Ã¢Å â€ Ã°ÂÂÂµ

Ã¢â€“Âª If every element of A is in B but they are not equal, then A is a Proper

Subset of B: Ã°ÂÂÂ´ Ã¢Å â€š Ã°ÂÂÂµ

Ã¢â€“Âª A = {1, 4, 7}

B = {1, 2, 3, 4, 5, 6, 7}

Ã°ÂÂÂ´ Ã¢Å â€š Ã°ÂÂÂµ : Proper Subset

Ã¢â€“ÂªA = {1, 4, 7}

Ã°ÂÂÂ´Ã¢Å â€ Ã°ÂÂÂµ

B = {1, 4, 7}

Subset

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Not a Subset

If just one element in set A does not exist in set B, then A is Not

a subset of B:

Ã°ÂÂÂ´ Ã¢Å â€ž Ã°ÂÂÂµ

Ã¢Å¾Â¢ Example:

If A = {1, 7, 9} and B = {1, 4, 5, 6, 7},

then Ã°ÂÂÂ´ Ã¢Å â€ž Ã°ÂÂÂµ

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Notes

Ã¢â€“Âª Any set (A) is a subset of itself.

Ã°ÂÂÂ´ Ã¢Å â€ Ã°ÂÂÂ´

Ã¢â€“Âª Every set (A) is a proper subset of the Universal set (U).

Ã°ÂÂÂ´ Ã¢Å â€šÃ°Ââ€˜Ë†

Ã¢â€“ÂªThe empty set ÃƒËœ is also a proper subset of every set (A).

Ã¢Ë†â€¦ Ã¢Å â€šÃ°ÂÂÂ´

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Subset

{3, 4, 5, 6} Ã¯Æ’Â {3, 4, 5, 6, 8}

{1, 2, 6}

GMTH 141A DR. NEMA SALEM

Ã¯Æ’Â {2, 4, 6, 8}

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Subsets and Proper Subsets: Exercises

Find the relation between sets.

1. A is the set of people living in Phoenix, Arizona, and USA.

B is the set of people living in Arizona and USA

2. C is the set of weekdays

D is the set of Monday, Tuesday, Wednesday, Thursday and Friday

3. A = {1, 2, 3, 5, 8, 9}

B={2, 3, 8, 11, 12, 13}

c={1, 2, 8}

4. A= {a, b, c, d, e}

B={b, c, d}

C={a, d, e, b, c}

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Examples

Answer: a) 10 b) 0, 10 c) -10, 0, -5, 10 d) all except square root of 5 e) Square root of 5 f) all of them

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Venn Diagram

Some elements are common

between A and B

U

U

A

B

All elements of A are

elements in B

U

Disjoint sets:

No common elements

U

U

A

B

A

B

A

B

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Set Operations/Algebra: Union

Ã¢â€“Âª Union of two sets A and B (A U B) is the set of all the elements in A or

B.

U

Ã¢â€“Âª A = {2 , 3 , 5 , 7 }

Ã¢â€“Âª B = {10 , 3 , 5 , 1}

A

B

Ã¢â€“Âª A U B = {1 , 2 , 3 , 5 , 7 , 10 }

GMTH 141A DR. NEMA SALEM

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Set Operations/Algebra: Intersection

Ã¢Å¾Â¢Intersection of two sets A and B (A Ã¢Ë†Â© B) is the set of all the elements

in A and B.

UU

Ã¢â€“ÂªA = {2,3,5,7}

Ã¢â€“Âª B = {10,3,5,1}

AA

B

Ã¢â€“Âª A Ã¢Ë†Â© B = {3,5}

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Set Operations/Algebra: Complement

Ã¢â€“ÂªComplement of a set A (Ac) is the set of all the elements which belong to U but

do not belong to A.

Ã¢â€“Âª U = {1,2,3,4,5,6,7,8,9,10}

U

Ã¢â€“Âª A = {3,5,1}

Ã¢â€“Âª Ac = {2,4,6,7,8,9,10}

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Set Operations/Algebra: Difference

Ã¢â€“Âª Set Difference of two sets A and B Ã¢â‚¬Å“(A – B)Ã¢â‚¬Â is the set of all the elements that

belong to A but not to B.

Ã¢â€“Âª A= {3,4,5,6,7}

U

Ã¢â€“Âª B = {3,5,1}

Ã¢â€“Âª A – B = {4,6,7}

A

GMTH 141A DR. NEMA SALEM

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B

25

Example

If A = {1,2,3,4,5,6,7} and B

= {6,7} are two sets.

Then, the difference of set

A and set B is given by;

A Ã¢â‚¬â€œ B = {1,2,3,4,5}

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Example

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Examples

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Set Properties

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DR. NEMA SALEM

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DeMorganÃ¢â‚¬â„¢s Law

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/TB

U

Consider the following Venn Diagram :

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DR. NEMA SALEM

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DeMorganÃ¢â‚¬â„¢s Law

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/TB

GMTH 141A

DR. NEMA SALEM

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Exercises

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Sketch Venn diagram for:

1. U={1, 2, 3, 4, 5}

Ã¢â€”Â¦ A={1, 2, 3}

Ã¢â€”Â¦ AC= Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦.

2. U={1, 2, 3, 4, 5, 6}

A={1, 2, 3}

B= Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦. As disjointed sets

3. U={1, 2, 3, 4, 5, 6}

A={1, 2, 3, 4}

B= Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦. As Proper sets

4. U={1, 2, 3, , 5, 6,7,8,10}

X = {1, 2, 6, 7}

Y={1, 3, 4, 5, 8}

X U Y = Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦.

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5. U={1, 2, 3, 5, 6,7,8,9,10}

X = {1, 6, 9}

Y={1, 3, 5, 6, 8, 9}

X U Y = Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦.

6. U={1, 2, 3, 4, 5}

A = {1, 2,3}

B = Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦..equal subsets

7. U={1, 2, 3, 4, 5, 6}

A = {2,3,4}

B={3,4,5}

A Ã¯Æ’â€¡ B= Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦

A Ã¯Æ’Ë† B =Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦Ã¢â‚¬Â¦.

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8. U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11},

A = {2, 3, 4, 6, 7, 8, 10}

B = {1, 3, 5, 7, 9}

C = {2, 3, 4}

9. Sketch Venn diagram and show the relations for:

oU is the set of whole numbers from 1 till 15

Ã¢â€”Â¦ A is the set of multiple of 3 within U

Ã¢â€”Â¦ B is the set of prime numbers within U

Ã¢â€”Â¦ C is the set of odd numbers within U

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10. Given Venn diagram, evaluate each of the following relations

a. 1. AÃ¯Æ’Ë†B

b. 2. AÃ¯Æ’â€¡B

U

c. 3. (AÃ¯Æ’Ë†B)C

d. 4. ACÃ¯Æ’â€¡B

e. 5. AÃ¯Æ’Ë†BC

B

20

A

2

6

7

21

4

5

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10

17

19

38

GMTH 141A

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Ã¢Ââ€˜List all the subsets of the set Q = {x, y, z}

Ã¢Å“â€œAnswer: The subsets of Q are { }, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}

and {x, y, z}

Ã¢Ââ€˜Q = {x, y, z}. How many subsets will Q have?

Ã¢Å“â€œAnswer: Number of subsets = 23 = 8

Ã¢Ââ€˜Draw a Venn diagram to represent the relationship between the sets. A = {1,

3, 5} and B = {1, 2, 3, 4, 5}

Ã¢Å“â€œAnswer:

GMTH 141A

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Let P = {3, 5, 7, 9, 11}

Q = {9, 11, 13} R = {3, 5, 9} and S = {13, 11}

Write Yes or No for the following:

(a) R Ã¢Å â€š P

(b) Q Ã¢Å â€š P

(c) R Ã¢Å â€š S

(d) S Ã¢Å â€š Q

(e) S Ã¢Å â€š P

(f) P Ã¢Å â€ž Q

(g) Q Ã¢Å â€ž R

(h) S Ã¢Å â€ž Q

Answer

Write all the subsets for the following.

(a) {3}

(b) {6, 11}

Answer: (a) Ã¢Ë†â€¦ , {3}

(c) Ã¢Ë†â€¦

(b) Ã¢Ë†â€¦, {6}, {11}, {6,11} c) Ã¢Ë†â€¦

GMTH 141A

DR. NEMA SALEM

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GMTH 141A

DR. NEMA SALEM

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Shade the following in Venn diagram

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Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, and C = {2, 3, 5, 7} three subsets of the

universal set U = {1, 2, 3, 4, 5, 6, 7, 8}.

Draw Venn diagram to represent the subsets A, B, C and the Universal set U.

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Handout #

Mathematics for Architecture and

Design Ã¢â‚¬â€œ GMTH 141A

Chapter Review R1

Geometry

1

Geometry

Points, lines, surfaces, solids

GMTH

BUSINESS

141A

INFOGRAPHIC

Math

for Arch

& Design

Perimeter and Area

Volume

Composite Figures

Solid Figures

2

Geometry is concerned with the various aspects of size, shape

and space.

In this unit, you will explore the concepts of angles, shapes,

area, and volume.

3

/tb

3

Points

A point is a location in space. It has no size.

It is named by a capital letter.

.

Read it as: Ã¢â‚¬Å“Point AÃ¢â‚¬Â

Write it as: Ã¢â‚¬Å“AÃ¢â‚¬Â

A

4

Lines

A line is a series of points that extends in two opposing

directions without end. Lines have no thickness.

Q

S

Read it as: Ã¢â‚¬Å“Line QSÃ¢â‚¬Â or Ã¢â‚¬Å“Line SQÃ¢â‚¬Â

Write it as: QS or SQ

5

Segments

A segment is a part of a line with two endpoints.

R

B

Read it as: Ã¢â‚¬Å“Segment RBÃ¢â‚¬Â or Ã¢â‚¬Å“Segment BRÃ¢â‚¬Â

Write it as:

RB

or

BR

6

Rays

A ray consists of an endpoint and all the points of a line on one

side of the endpoint.

C

D

Read it as: Ã¢â‚¬Å“Ray CDÃ¢â‚¬Â (the order does matter)

Write it as:

CD

7

J

Parallel Lines- lines that never cross or become

further apart from each other.

L

JK Ã¢â€¢â€˜ LM

K

M

Read: Ã¢â‚¬Å“Line JK is parallel to

line LMÃ¢â‚¬Â

8

Intersecting Lines- lines that cross each other.

O

Q

OP intersects

R

P

QR

9

Collinear and Non-collinear Points

Ã¢â‚¬Â¢!”#$%&’%()%’)*+'”$’%(+’&),+’-#$+’)*+’.)–+/’.”–#$+)*’0″#$%&1

Ã¢â‚¬Â¢23′)’&#$4-+’-#$+’.)$$”%’5+’/*)6$’%(*”74(‘)–‘%(+’0″#$%&8’%(+$’%(+’

0″#$%&’)*+’$”$9.”–#$+)*1

C D E F

Collinear points

U

V

W

X

Noncollinear points

10

Planes

A plane is a flat surface with no thickness that extends

without end in all directions.

Intersecting lines are lines that cross

(intersect) at exactly one point.

Parallel lines are lines that do not cross (they

have no points in common).

Skew lines are not parallel and they do not

intersect. They lie in different planes.

11

Handout #

Perimeter and Area

12

Handout #

Volume

GMTH 141-A Ã¢â‚¬â€œ Mathematics for Architecture & Design

13

/tb

13

Handout #

Most Common Types of Shapes

14

/tb

14

Handout #

15

Handout #

16

/tb

16

Handout #

17

Handout #

18

Handout #

19

/tb

19

Handout #

Cylindrical

Design

20

/tb

20

Handout #

Pyramids

Design

21

/tb

21

Dimensions of length, area and

volume

length

area

volume

length

mm, cm, m, km, inch,

foot

two dimensions

length Ãƒâ€”

length

mm2, cm2, m2, hectare,

km2, square inch,

square foot

three

dimensions

length Ãƒâ€”

length Ãƒâ€”

length

mm3, cm3, m3, km2,

cubic inch, cubic foot

one dimension

22

Rectangles

The perimeter of a rectangle with length l and width w can be written as:

l

Perimeter = 2l + 2w

or

w

Perimeter = 2(l + w)

The area of a rectangle is given as:

Area = lw

23

Squares

When the length and the width of a rectangle are equal we call it a square. A

square is just a special type of rectangle.

The perimeter of a square with length l is given as:

Perimeter = 4l

l

The area of a square is given as:

Area = l2

24

Shapes

made

from

rectangles

How can we find the area of the shaded shape?

9 cm

4 cm

A

We can think of this shape

as being made up of one

rectangle cut out of another

rectangle.

Label the rectangles A and B.

11 cm

B

6 cm

Area A = 9 Ãƒâ€” 11 = 99 cm2

Area B = 4 Ãƒâ€” 6 = 24 cm2

Total area = 99 Ã¢â‚¬â€œ 24 = 75 cm2

25

The area of a triangle

h

b

The area of a triangle with base b and perpendicular height h is given by:

1

Area of a triangle =

bh

2

26

The area of a triangle

h

b

Any side of the triangle can be taken as the base, as long as the height is

perpendicular to it:

b

h

b

h

27

The area of a triangle

Suppose that instead of the height of a triangle, we are given

the base, one of the sides and the included angle.

We can use trigonometry to find the area of the triangle:

A

c

B

b

a

Area of triangle ABC =

C

1

ab sin C

2

28

The area of a parallelogram

The area of any parallelogram can be found using the formula:

Area of a parallelogram = base Ãƒâ€” perpendicular height

perpendicular

height

base

Or using letter symbols,

A = bh

29

The area of a parallelogram

What is the area of this parallelogram?

4.9 cm

4.5 cm

We can ignore

this length

7 cm

Area of a parallelogram = bh

= 7 Ãƒâ€” 4.5

= 31.5 cm2

30

The area of a trapezium

The area of any trapezium can be found using the formula:

Area of a trapezium =

a

1

(sum of parallel sides) Ãƒâ€” height

2

perpendicular

height

b

Or using letter symbols,

A=

1

(a + b)h

2

31

The area of a trapezium

What is the area of this trapezium?

1

Area of a trapezium =

(a + b)h

2

6m

9m

=

1

(6 + 14) Ãƒâ€” 9

2

1

=

Ãƒâ€” 20 Ãƒâ€” 9

2

14 m

= 90 m2

32

The area of a trapezium

What is the area of this trapezium?

1

Area of a trapezium =

(a + b)h

2

=

9m

4m

12 m

1

(9 + 4) Ãƒâ€” 12

2

1

=

Ãƒâ€” 13 Ãƒâ€” 12

2

= 78 m2

33

Cubes and cuboids

A cuboid is a 3-D shape with edges of different lengths. All of its

faces are rectangular or square.

How many faces does a cuboid have?

6

Face

How many edges does a cuboid have?

12

Edge

Vertex

How many vertices does a cuboid

8

have?

A cube is a special type of cuboid with edges of equal length. All of its faces are

square.

34

Length around the edges

Suppose we have a cuboid of length 5 cm, width 4 cm and height 3 cm. What

is the total length around the edges?

Imagine the cuboid as a hollow wire frame:

The cuboid has 12 edges.

3 cm

4 edges are 5 cm long.

4 edges are 4 cm long.

4 cm

4 edges are 3 cm long.

5 cm

Total length around the edges = 4 Ãƒâ€” 5 + 4 Ãƒâ€” 4 + 4 Ãƒâ€” 3

= 20 + 16 + 12

= 48 cm

35

Length around the edges

To find the length around the edges of a cuboid of length l, width w and height h we can use the formula:

Length around the edges = 4l + 4w + 4h

or

Length around the edges = 4(l + w + h)

To find the length around the edges of a cube of length l we can use the formula:

Length around the edges = 12l

36

Surface area of a cuboid

To find the surface area of a cuboid, we calculate the total area of all of the faces.

A cuboid has 6 faces.

The top and the bottom of the cuboid have

the same area.

37

Surface area of a cuboid

To find the surface area of a cuboid, we calculate the total area of all of the faces.

A cuboid has 6 faces.

The front and the back of the cuboid have

the same area.

38

Surface area of a cuboid

To find the surface area of a cuboid, we calculate the total area of all of the faces.

A cuboid has 6 faces.

The left hand side and the right hand side

of the cuboid have the same area.

39

Formula for the surface area of a cuboid

We can find the formula for the surface area of a cuboid as follows.

l

h

w

Surface area of a cuboid =

2 Ãƒâ€” lw

Top and bottom

+ 2 Ãƒâ€” hw

Front and back

+ 2 Ãƒâ€” lh

Left and right side

Surface area of a cuboid = 2lw + 2hw + 2lh

40

Surface area of a cube

How can we find the surface area of a cube of length l?

All six faces of a cube have the

same area.

The area of each face is l Ãƒâ€” l = l2

Therefore,

l

Surface area of a cube = 6l2

41

Volume of a cuboid

The following cuboid is made out of interlocking cubes.

How many cubes does it contain?

42

Volume of a cuboid

We can work this out by dividing the cuboid into layers.

The number of cubes in each layer

can be found by multiplying the

number of cubes along the length

by the number of cubes along the

width.

3 Ãƒâ€” 4 = 12 cubes in each layer

There are three layers altogether

so the total number of cubes in the

cuboid = 3 Ãƒâ€” 12 = 36 cubes

43

Volume of a cuboid

We can find the volume of a cuboid by multiplying the area of the base by the height.

The area of the base

= length Ãƒâ€” width

So,

height, h

Volume of a cuboid

= length Ãƒâ€” width Ãƒâ€” height

width, w

length, l

= lwh

44

Volume of a cube

How can we find the volume of a cube of length l?

The length, width and height of a

cube are the same.

Therefore:

Volume of a cube

l

= (length of one edge)3

= l3

45

Volume of shapes made from cuboids

What is the volume of this 3-D shape?

3 cm

We can think of this shape as

two cuboids joined together.

3 cm

4 cm

Volume of the green cuboid

= 6 Ãƒâ€” 3 Ãƒâ€” 3 = 54 cm3

6 cm

Volume of the blue cuboid

= 3 Ãƒâ€” 2 Ãƒâ€” 2 = 12 cm3

Total volume

5 cm

= 54 + 12 = 66 cm3

46

Prisms

A prism is a 3-D shape that has a constant cross-section along its length.

For example, this hexagonal prism has the same hexagonal

cross-section throughout its length.

This is called a

hexagonal prism

because its crosssection is a hexagon.

47

Volume of a prism The volume of a prism is found by multiplying the area of its

cross-section A by its length l (or by its height if it is

standing on its cross-section).

l

h

A

A

V = Al

or

V = Ah

48

Volume of a prism

What is the volume of this triangular prism?

7.2 cm

4 cm

5 cm

Area of cross-section = Ã‚Â½ Ãƒâ€” 5 Ãƒâ€” 4 = 10 cm2

Volume of prism = 10 Ãƒâ€” 7.2 = 72 cm3

49

Volume of a prism

What is the volume of this prism?

12 m

7m

4m

3m

5m

Area of cross-section = 7 Ãƒâ€” 12 Ã¢â‚¬â€œ 4 Ãƒâ€” 3 = 84 Ã¢â‚¬â€œ 12 = 72 m2

Volume of prism = 72 Ãƒâ€” 5 = 360 m3

50

Surface area of a prism

Here is the net of a triangular prism.

What is its surface area?

10 cm

13 cm

260

60

200

12 cm

260

20 cm

We can work out the area

of each face and write it in

the diagram of the net.

60

Total surface area

= 60 + 60 + 200 + 260 + 260

= 840 cm2

51

Pyramids

A pyramid is a 3-D shape whose base is usually a polygon

but can also be a shape with curved edges. The faces

rising up from the base meet at a common vertex or apex.

The most common pyramids are:

A tetrahedron

or triangular

pyramid.

A square-based

pyramid

A cone

52

Volume of a pyramid

The volume of a pyramid is found by multiplying the area of its base A by its perpendicular height h and

dividing by 3.

Apex

slant height

h

A

base

Volume of a pyramid =

V=

1

3

1

3

Ãƒâ€” area of base Ãƒâ€” height

Ah

53

Volume of a pyramid

What is the volume of this rectangle-based pyramid?

Area of the base = 5 Ãƒâ€” 3

= 15 cm2

8 cm

Volume of pyramid =

3 cm

5 cm

=

1

3

1

3

Ah

Ãƒâ€” 15 Ãƒâ€” 8

= 40 cm3

54

Surface area of a pyramid

Here is the net of a regular tetrahedron.

What is its surface area?

Area of each face = Ã‚Â½bh

= Ã‚Â½ Ãƒâ€” 6 Ãƒâ€” 5.2

= 15.6 cm2

5.2 cm

Surface area = 4 Ãƒâ€” 15.6

= 62.4 cm2

6 cm

55

Volume of a cylinder

A cylinder is a special type of prism with a circular cross-section.

Remember, the volume of a prism can be found by multiplying the area of the

cross-section by the height of the prism.

r

The volume of a cylinder is given by:

Volume = area of circular base Ãƒâ€” height

h

or

V = Ãâ‚¬r2h

56

Surface area of a cylinder

To find the formula for the surface area of a cylinder we can draw its net.

r

h

?

2Ãâ‚¬r

How can we find the width of the curved face?

The width of the curved face is equal to the

circumference of the circular base, 2Ãâ‚¬r.

Area of curved face = 2Ãâ‚¬rh

Area of 2 circular faces = 2 Ãƒâ€” Ãâ‚¬r2

Surface area of a cylinder = 2Ãâ‚¬rh + 2Ãâ‚¬r2

or

Surface area = 2Ãâ‚¬r(h + r)

57

Volume of a cone

A cone is a special type of pyramid with a circular base.

Remember, the volume of a pyramid can be found by multiplying the

area of the base by the height and dividing by 3.

The volume of a cone is given by:

Volume =

h

1

3

Ãƒâ€” area of circular base Ãƒâ€” height

or

r

V=

1

3

Ãâ‚¬r2h

58

Volume and surface area of a sphere

A sphere is a 3-D shape whose surface is always the same distance from the

center. This fixed distance is the radius of the sphere.

For a sphere of radius r :

r

Volume =

4

3

Ãâ‚¬r3

and

Surface area = 4Ãâ‚¬r2

59

Most Common Types of Angles

n

n

The most common unit for measuring angles is the degree.

The major types of angles are acute angle, right angle, obtuse angle

and straight angle.

60

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60

Types of Triangles: Angles

n

Note: The sum of the measures of the angles of any triangle is 180Ã‚Â°.

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61

Polygons

62

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Handout #

Sept. 4!

Composite Figures

!”#$%&$'()* +(,-.*”(‘”+$.%*/”+.$%”)0$”$.”%$.*”+(,-.*’1

2$”+(3/”)4*”5.*5″$+”5″#$%&$'()*”+(,-.*6

7(3/”)4*”5.*5′”$+”*5#4″+(,-.*”)4*3″5// )4*%”-&1

2$”+(3/”)4*”5.*5″$+”5″’45/*/”.*,($38″9$-“3**/”)$”‘-:).5#) )4*”5.*5’1

Area of square:

A = lw = 6(6) = 36 ft2

Area of circle:

A = pr2

A = p(3)2 = 9p ft2

Area of semicircle =

Ã‚Â½ (9p) = 4.5p ft2

Total area of figure:

Add areas of square and

semicircle:

A = 36 + 4.5p cm2

46

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63

Handout #

Area of Composite Figure

6cm

8cm

12cm

8cm

–

4cm

4cm

64

Handout #

12cm

6cm

8cm

+

8cm

4cm

1

4cm

A = bh

= 12 Ã‚Â´ 8

= 96 cm 2

1

A = (b1 + b2 )h

2

1

= Ã‚Â´ ( 6 + 4) Ã‚Â´ 8

2

= 40cm 2

4cm

bh

A=

2

4Ã‚Â´ 4

=

2

= 8cm 2

Total Area = 96+40-8= 128cm2

65

Handout #

Total Area

66

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Handout #

Shaded Area

67

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Handout #

Shaded Area

68

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Handout #

Area Shaded

69

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Handout #

Area Shaded

70

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70

Handout #

71

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71

Handout #

Surface Area

72

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Handout #

Surface Area

73

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73

Handout #

Volume of Prisms

74

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Handout #

Volume of Prisms

75

Handout #

Composite Solid Figures

76

Handout #

Composite Solid Figures

77

Handout #

Composite Solid Figures

78

Handout #

Composite Solid Figures

GMTH 141-A Ã¢â‚¬â€œ Mathematics for Architecture & Design

79

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79

Handout #

Pyramid & Cone

80

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Handout #

Prism

81

Handout #

Volume of Pyramids

82

Handout #

Volume of Cylinders

83

Handout #

Volume of a Cone

84

Handout #

Find the surface area of the sphere

S = 4pr2

Surface area of a sphere

= 4p(4.5)2

Replace r with 4.5.

Ã¢â€°Ë† 254.5

Simplify.

85

Handout #

Example

A. Find the surface area of the hemisphere.

Find half the area of a sphere with the radius of

3.7 millimeters. Then add the area of the great circle.

GMTH 141-A Ã¢â‚¬â€œ Mathematics for Architecture & Design

86

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86

Handout #

Surface area of a

hemisphere

Replace r with 3.7.

Ã¢â€°Ë† 129.0

Use a calculator.

Answer: about 129.0 mm2

87

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