UNIT 3 CIRCLES AND VOLUME Lesson 5: Explaining and Applying Area and Volume Formulas Instruction
|
|
- Reginald Curtis Farmer
- 5 years ago
- Views:
Transcription
1 Prerequisite Skills This lesson requires the use of the following skills: using formulas for the surface areas of polygons and circles performing calculations with the angles in circles using the Pythagorean Theorem using ratios of trigonometry understanding how to bisect angles and side lengths Introduction You have used the formulas for finding the circumference and area of a circle. In this lesson, you will prove why the formulas for circumference and area work. You will see how the ratio of π can be proven. Key Concepts You know that the circumference of a specific circle divided by its diameter is the ratio pi, written as π. Pi (π) is an irrational number that cannot be written as a repeating decimal or as a fraction. It has an infinite number of non-repeating decimal places. We know that the circumference of a circle = π diameter or π radius. Therefore, π = circumference diameter = circumference radius Long ago, mathematicians didn t yet know the value of pi. Archimedes, a great mathematician from ancient Greece, used inscribed polygons to determine the value of pi. He started by inscribing a regular hexagon in a circle.. U3-03
2 A B G C D F E He determined that each side of the hexagon equals the radius of the circle. AB = BD = DE = EF = FG = GA = CE Archimedes realized that if the perimeter of the hexagon were equal to the circumference of the circle, then both would equal 6r. This would mean that π = 3. However, the circumference is larger than the hexagon; therefore, Archimedes thought, π must be larger than 3. Next, Archimedes inscribed a regular dodecagon a 1-sided polygon in the circle. The perimeter of the dodecagon was much closer to the actual perimeter of the circle. U3-04
3 He calculated the perimeter of the dodecagon to be approximately This means π However, the circumference of the circle is still larger than the dodecagon, so π must be greater than Next, Archimedes inscribed a 4-sided regular polygon and calculated its perimeter. This polygon s perimeter is even closer to the circumference of a circle. Archimedes found that the ratio of the perimeter to the diameter is closer to the value of π. Archimedes kept going with this process until he had inscribed a 48-sided polygon. As the number of sides of a polygon increases, the polygon looks more and more like a circle. As he worked, the number for the ratio of π became more and more accurate. The more sides an inscribed polygon has, the closer its perimeter is to the actual circumference of the circle. Therefore, Archimedes determined that as the number of sides of a polygon inside a circle increases, the calculation approaches the limit for the value of π. A limit is the value that a sequence approaches as a calculation becomes more and more accurate. This limit cannot be reached. Theoretically, if the polygon had an infinite number of sides, π could be calculated. This is the basis for the formula for finding the circumference of a circle. Increasing the number of side lengths for the inscribed polygon causes the polygon s perimeter to get closer and closer to the length of the circumference of the circle. The area of the circle can be derived similarly using dissection principles. Dissection involves breaking a figure down into its components. In the diagram that follows, a circle has been divided into four equal sections. If you cut the four sections from the circle apart, you can arrange them to resemble a rectangle. r r U3-05
4 The width of the rectangle equals the radius, r, of the original circle. The length is equal to half of the circumference, or π r. The circle in the diagram below has been divided into 16 equal sections. You can arrange the 16 segments to form a new rectangle. This figure looks more like a rectangle. r r As the number of sections increases, the rounded bumps along its length and the slant of its width become less and less distinct. The figure will approach the limit of being a rectangle. The formula for the area of a rectangle is l w = a. The length of the rectangle made out of the circle segments is π r. The width is r. Thus, the area of the circle is a = r π r = π r. This proof is a dissection of the circle. Remember that a sector is the part of a circle that is enclosed by a central angle. A central angle has its vertex on the center of the circle. A sector will have an angular measure greater than 0º and less than 360º. Common Errors/Misconceptions not realizing that there is more than one way to prove a formula using the wrong formula for area, circumference, or the area of a sector using the diameter in a formula instead of the radius U3-06
5 Guided Practice Example 1 Show how the perimeter of a hexagon can be used to find an estimate for the circumference of a circle that has a radius of 5 meters. Compare the estimate with the circle s perimeter found by using the formula C = πr. 1. Draw a circle and inscribe a regular hexagon in the circle. Find the length of one side of the hexagon and multiply that length by 6 to find the hexagon s perimeter.. Create a triangle with a vertex at the center of the circle. Draw two line segments from the center of the circle to vertices that are next to each other on the hexagon. C B P U3-07
6 3. To find the length of BC, first determine the known lengths of PB and PC. Both lengths are equal to the radius of circle P, 5 meters. 4. Determine m CPB. The hexagon has 6 sides. A central angle drawn from P will be equal to one-sixth of the number of degrees in circle P. 1 m CPB = 360 = 60 6 The measure of CPB is 60º. 5. Use trigonometry to find the length of BC. Make a right triangle inside of PBC by drawing a perpendicular line, or altitude, from P to BC. C D B P U3-08
7 6. Determine m BPD. DP bisects, or cuts in half, CPB. Since the measure of CPB was found to be 60º, divide 60 by to determine m BPD. 60 = 30 The measure of BPD is 30º. 7. Use trigonometry to find the length of BD and multiply that value by to find the length of BC. BD is opposite BPD. The length of the hypotenuse, PB, is 5 meters. The trigonometry ratio that uses the opposite and hypotenuse lengths is sine. BD sinbpd = sin 30º = = BD Substitute the sine of 30º = BD Multiply both sides of the equation by 5. BD =.5 The length of BD is.5 meters. Since BC is twice the length of BD, multiply.5 by. BC =.5 = 5 The length of BC is 5 meters. U3-09
8 8. Find the perimeter of the hexagon. Perimeter = BC 6 = 5 6 = 30 The perimeter of the hexagon is 30 meters. 9. Compare the estimate with the calculated circumference of the circle. Calculate the circumference. C = πr Formula for circumference C = π 5 Substitute 5 for r. C meters Find the difference between the perimeter of the hexagon and the circumference of the circle = meters The formula for circumference gives a calculation that is meters longer than the perimeter of the hexagon. You can show this as a percentage difference between the two values = = 451. % From a proportional perspective, the circumference calculation is approximately 4.51% larger than the estimate that came from using the perimeter of the hexagon. If you inscribed a regular polygon with more side lengths than a hexagon, the perimeter of the polygon would be closer in value to the circumference of the circle. U3-10
9 Example Show how the area of a hexagon can be used to find an estimate for the area of a circle that has a radius of 5 meters. Compare the estimate with the circle s area found by using the formula A = π r. 1. Inscribe a hexagon into a circle and divide it into 6 equal triangles.. Use the measurements from Example 1 to find the area of one of the six triangles. C D B P First, determine the formula to use. A= 1 bh A PBC = 1 BC PD Area formula for a triangle Rewrite the formula with the base and height of the triangle whose area you are trying to determine. (continued) U3-11
10 From Example 1, the following information is known: BC = 5 meters BD =.5 meters m BPD =30º You need to find the height, PD. In BPD, the height, PD, is the adjacent side length. Since the hypotenuse, BP, is a radius of the circle, it is 5 meters. Since the measure of BPD and the hypotenuse are known, use the cosine of 30º to find PD. cos30º = PD = PD Substitute the cosine of 30º = PD Multiply both sides by 5. PD meters Now that the length of PD is known, use that information to find the area of PBC using the formula determined earlier. A PBC = 1 BC PD Area formula for PBC A PBC A PBC = ( ) ( ) m Substitute the values of BC and PD. 3. Find the area of the hexagon. Multiply the area of one triangle times 6, the number of triangles in the hexagon = m The area of hexagon is about m. U3-1
11 4. Compare the area of the hexagon with the area of the circle. Find the area of the circle. Acircle P =πr Formula for the area of a circle A circle P =π 5 Substitute the value for the radius. A circle P m The actual area of circle P is about m. Find the difference between the area of the hexagon and the area of the circle = The actual area of the circle is approximately m greater than the hexagon s area. Show the difference as a percentage % The actual area of the circle is about 17.30% larger than the estimate found by using the area of the hexagon. The estimate of a circle s area calculated by using an inscribed polygon can be made closer to the actual area of the circle by increasing the number of side lengths of the polygon. U3-13
12 Example 3 Find the area of a circle that has a circumference of 100 meters. 1. First, find the measure of the radius by using the formula for circumference. C = πr 100 = πr r = 100 π r m. Calculate the area by using the formula for the area of a circle. A = π r A π( ) A π m A m The area of a circle with a circumference of 100 meters is approximately 796 m. U3-14
13 Example 4 What is the circumference of a circle that has an area of 1,000 m? 1. First, find the radius by solving for r in the formula for the area of the circle. A = π r Formula for the area of a circle 1000 = π r Substitute A into the equation = r π Divide both sides by π. r Simplify. r =± Take the square root of both sides. Use only the positive result, as distance is r m always positive. The radius is approximately meters.. Find the circumference using the formula C = π r. C = πr Formula for circumference of a circle C = π(17.841) Substitute for r. C 11.1 m The circumference of a circle with an area of 1,000 m is approximately 11.1 meters. U3-15
You know that the circumference of a specific circle divided by its diameter is the ratio pi, written as.
Unit 6, Lesson.1 Circumference and Area of a Circle You have used the formulas for finding the circumference and area of a circle. In this lesson, you will prove why the formulas for circumference and
More informationLength and Area. Charles Delman. April 20, 2010
Length and Area Charles Delman April 20, 2010 What is the length? Unit Solution Unit 5 (linear) units What is the length? Unit Solution Unit 5 2 = 2 1 2 (linear) units What is the perimeter of the shaded
More information1. AREAS. Geometry 199. A. Rectangle = base altitude = bh. B. Parallelogram = base altitude = bh. C. Rhombus = 1 product of the diagonals = 1 dd
Geometry 199 1. AREAS A. Rectangle = base altitude = bh Area = 40 B. Parallelogram = base altitude = bh Area = 40 Notice that the altitude is different from the side. It is always shorter than the second
More informationDay 5: Inscribing and Circumscribing Getting Closer to π: Inscribing and Circumscribing Polygons - Archimedes Method. Goals:
Day 5: Inscribing and Circumscribing Getting Closer to π: Inscribing and Circumscribing Polygons - Archimedes Method Goals: Construct an inscribed hexagon and dodecagon. Construct a circumscribed hexagon
More informationGeometry 10 and 11 Notes
Geometry 10 and 11 Notes Area and Volume Name Per Date 10.1 Area is the amount of space inside of a two dimensional object. When working with irregular shapes, we can find its area by breaking it up into
More informationUNIT 3 CIRCLES AND VOLUME Lesson 5: Explaining and Applying Area and Volume Formulas Instruction
Prerequisite Skills This lesson requires the use of the following skills: understanding and using formulas for the volume of prisms, cylinders, pyramids, and cones understanding and applying the formula
More informationLog1 Contest Round 2 Theta Circles, Parabolas and Polygons. 4 points each
Name: Units do not have to be included. 016 017 Log1 Contest Round Theta Circles, Parabolas and Polygons 4 points each 1 Find the value of x given that 8 x 30 Find the area of a triangle given that it
More informationGeometry B. The University of Texas at Austin Continuing & Innovative Education K 16 Education Center 1
Geometry B Credit By Exam This Credit By Exam can help you prepare for the exam by giving you an idea of what you need to study, review, and learn. To succeed, you should be thoroughly familiar with the
More informationPerimeter. Area. Surface Area. Volume. Circle (circumference) C = 2πr. Square. Rectangle. Triangle. Rectangle/Parallelogram A = bh
Perimeter Circle (circumference) C = 2πr Square P = 4s Rectangle P = 2b + 2h Area Circle A = πr Triangle A = bh Rectangle/Parallelogram A = bh Rhombus/Kite A = d d Trapezoid A = b + b h A area a apothem
More information10-2. Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Geometry
Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Find the unknown side lengths in each special right triangle. 1. a 30-60 -90 triangle with hypotenuse 2 ft 2. a 45-45 -90 triangle with leg length
More informationLesson Title 2: Problem TK Solving with Trigonometric Ratios
Part UNIT RIGHT solving TRIANGLE equations TRIGONOMETRY and inequalities Lesson Title : Problem TK Solving with Trigonometric Ratios Georgia Performance Standards MMG: Students will define and apply sine,
More information10.6 Area and Perimeter of Regular Polygons
10.6. Area and Perimeter of Regular Polygons www.ck12.org 10.6 Area and Perimeter of Regular Polygons Learning Objectives Calculate the area and perimeter of a regular polygon. Review Queue 1. What is
More informationUnit 4 End-of-Unit Assessment Study Guide
Circles Unit 4 End-of-Unit Assessment Study Guide Definitions Radius (r) = distance from the center of a circle to the circle s edge Diameter (d) = distance across a circle, from edge to edge, through
More informationProperties of a Circle Diagram Source:
Properties of a Circle Diagram Source: http://www.ricksmath.com/circles.html Definitions: Circumference (c): The perimeter of a circle is called its circumference Diameter (d): Any straight line drawn
More information11.1 Understanding Area
/6/05. Understanding rea Counting squares is neither the easiest or the best way to find the area of a region. Let s investigate how to find the areas of rectangles and squares Objective: fter studying
More informationNumber. Number. Number. Number
Order of operations: Brackets Give the order in which operations should be carried out. Indices Divide Multiply Add 1 Subtract 1 What are the first 10 square numbers? The first 10 square numbers are: 1,
More informationAngles. An angle is: the union of two rays having a common vertex.
Angles An angle is: the union of two rays having a common vertex. Angles can be measured in both degrees and radians. A circle of 360 in radian measure is equal to 2π radians. If you draw a circle with
More informationSyllabus Form 3. Objectives Students will be able to: Mathematics Department. Revision of numbers. Form 3 Syllabus Page 1 of 7
Syllabus Form 3 Topic Revision of numbers Content 1. Rounding 2. Estimation 3. Directed numbers 4. Working with decimals 5. Changing fractions to 6. decimals and vice versa 7. 7.1. The reciprocal of a
More informationGeometry: Traditional Pathway
GEOMETRY: CONGRUENCE G.CO Prove geometric theorems. Focus on validity of underlying reasoning while using variety of ways of writing proofs. G.CO.11 Prove theorems about parallelograms. Theorems include:
More informationACT Math test Plane Geometry Review
Plane geometry problems account for 14 questions on the ACT Math Test that s almost a quarter of the questions on the Subject Test. If you ve taken high school geometry, you ve probably covered all of
More information3. Radius of incenter, C. 4. The centroid is the point that corresponds to the center of gravity in a triangle. B
1. triangle that contains one side that has the same length as the diameter of its circumscribing circle must be a right triangle, which cannot be acute, obtuse, or equilateral. 2. 3. Radius of incenter,
More information8. T(3, 4) and W(2, 7) 9. C(5, 10) and D(6, -1)
Name: Period: Chapter 1: Essentials of Geometry In exercises 6-7, find the midpoint between the two points. 6. T(3, 9) and W(15, 5) 7. C(1, 4) and D(3, 2) In exercises 8-9, find the distance between the
More informationUnit 6: Connecting Algebra and Geometry Through Coordinates
Unit 6: Connecting Algebra and Geometry Through Coordinates The focus of this unit is to have students analyze and prove geometric properties by applying algebraic concepts and skills on a coordinate plane.
More informationACT SparkNotes Test Prep: Plane Geometry
ACT SparkNotes Test Prep: Plane Geometry Plane Geometry Plane geometry problems account for 14 questions on the ACT Math Test that s almost a quarter of the questions on the Subject Test If you ve taken
More informationGCSE Maths: Formulae you ll need to know not
GCSE Maths: Formulae you ll need to know As provided by AQA, these are the formulae required for the new GCSE These will not be given in the exam, so you will need to recall as well as use these formulae.
More informationChapter 10 Similarity
Chapter 10 Similarity Def: The ratio of the number a to the number b is the number. A proportion is an equality between ratios. a, b, c, and d are called the first, second, third, and fourth terms. The
More informationGEOMETRY SEMESTER 2 REVIEW PACKET 2016
GEOMETRY SEMESTER 2 REVIEW PACKET 2016 Your Geometry Final Exam will take place on Friday, May 27 th, 2016. Below is the list of review problems that will be due in order to prepare you: Assignment # Due
More informationMath-2 Lesson 6-3: Area of: Triangles, rectangles, circles and Surface Area of Pyramids
Math- Lesson 6-3: rea of: Triangles, rectangles, circles and Surface rea of Pyramids SM: Lesson 6-3 (rea) For the following geometric shapes, how would you answer the question; how big is it? Describe
More informationCounter argument against Archimedes theory
Counter argument against Archimedes theory Copyright 007 Mohammad-Reza Mehdinia All rights reserved. Contents 1. Explanations and Links. Page 1-. Facts: Page 3-4 3. Counter argument against Archimedes
More informationGeometry. Geometry is one of the most important topics of Quantitative Aptitude section.
Geometry Geometry is one of the most important topics of Quantitative Aptitude section. Lines and Angles Sum of the angles in a straight line is 180 Vertically opposite angles are always equal. If any
More informationHustle Geometry SOLUTIONS MAΘ National Convention 2018 Answers:
Hustle Geometry SOLUTIONS MAΘ National Convention 08 Answers:. 50.. 4. 8 4. 880 5. 6. 6 7 7. 800π 8. 6 9. 8 0. 58. 5.. 69 4. 0 5. 57 6. 66 7. 46 8. 6 9. 0.. 75. 00. 80 4. 8 5 5. 7 8 6+6 + or. Hustle Geometry
More informationPostulates, Theorems, and Corollaries. Chapter 1
Chapter 1 Post. 1-1-1 Through any two points there is exactly one line. Post. 1-1-2 Through any three noncollinear points there is exactly one plane containing them. Post. 1-1-3 If two points lie in a
More informationThe radius for a regular polygon is the same as the radius of the circumscribed circle.
Perimeter and Area The perimeter and area of geometric shapes are basic properties that we need to know. The more complex a shape is, the more complex the process can be in finding its perimeter and area.
More informationArea rectangles & parallelograms
Area rectangles & parallelograms Rectangles One way to describe the size of a room is by naming its dimensions. So a room that measures 12 ft. by 10 ft. could be described by saying its a 12 by 10 foot
More informationTo find the surface area of a pyramid and a cone
11-3 Surface Areas of Pyramids and Cones Common Core State Standards G-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects. MP 1, MP 3, MP 4, MP 6, MP 7 Objective To find
More informationMathematical derivations of some important formula in 2D-Geometry by HCR
From the SelectedWorks of Harish Chandra Rajpoot H.C. Rajpoot Summer March 31, 2018 Mathematical derivations of some important formula in 2D-Geometry by HCR Harish Chandra Rajpoot, HCR Available at: https://works.bepress.com/harishchandrarajpoot_hcrajpoot/61/
More informationDEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.
DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent
More informationMATHEMATICS FOR ENGINEERING TRIGONOMETRY
MATHEMATICS FOR ENGINEERING TRIGONOMETRY TUTORIAL SOME MORE RULES OF TRIGONOMETRY This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves
More informationStudy Guide and Review
Choose the term that best matches the statement or phrase. a square of a whole number A perfect square is a square of a whole number. a triangle with no congruent sides A scalene triangle has no congruent
More informationThis unit is built upon your knowledge and understanding of the right triangle trigonometric ratios. A memory aid that is often used was SOHCAHTOA.
Angular Rotations This unit is built upon your knowledge and understanding of the right triangle trigonometric ratios. A memory aid that is often used was SOHCAHTOA. sin x = opposite hypotenuse cosx =
More informationMeasurement 1 PYTHAGOREAN THEOREM. The area of the square on the hypotenuse of a right triangle is equal to the sum of the areas of
Measurement 1 PYTHAGOREAN THEOREM Remember the Pythagorean Theorem: The area of the square on the hypotenuse of a right triangle is equal to the sum of the areas of the squares on the other two sides.
More informationAngles. Classification Acute Right Obtuse. Complementary s 2 s whose sum is 90 Supplementary s 2 s whose sum is 180. Angle Addition Postulate
ngles Classification cute Right Obtuse Complementary s 2 s whose sum is 90 Supplementary s 2 s whose sum is 180 ngle ddition Postulate If the exterior sides of two adj s lie in a line, they are supplementary
More informationMathematics Department Inverclyde Academy
Common Factors I can gather like terms together correctly. I can substitute letters for values and evaluate expressions. I can multiply a bracket by a number. I can use common factor to factorise a sum
More informationAppendix E. Plane Geometry
Appendix E Plane Geometry A. Circle A circle is defined as a closed plane curve every point of which is equidistant from a fixed point within the curve. Figure E-1. Circle components. 1. Pi In mathematics,
More informationMoore Catholic High School Math Department
Moore Catholic High School Math Department Geometry Vocabulary The following is a list of terms and properties which are necessary for success in a Geometry class. You will be tested on these terms during
More informationCommon Core Cluster. Experiment with transformations in the plane. Unpacking What does this standard mean that a student will know and be able to do?
Congruence G.CO Experiment with transformations in the plane. G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point,
More informationKillingly Public Schools. Grades Draft Sept. 2002
Killingly Public Schools Grades 10-12 Draft Sept. 2002 ESSENTIALS OF GEOMETRY Grades 10-12 Language of Plane Geometry CONTENT STANDARD 10-12 EG 1: The student will use the properties of points, lines,
More information11.4 CIRCUMFERENCE AND ARC LENGTH 11.5 AREA OF A CIRCLE & SECTORS
11.4 CIRCUMFERENCE AND ARC LENGTH 11.5 AREA OF A CIRCLE & SECTORS Section 4.1, Figure 4.2, Standard Position of an Angle, pg. 248 Measuring Angles The measure of an angle is determined by the amount of
More informationCARIBBEAN CORRESPONDENCE SCHOOL
Final Examination CARIBBEAN CORRESPONDENCE SCHOOL Module Name: Groups: Duration: MATHEMATICS Online 3 Hours INSTRUCTIONS TO CANDIDATES 1. This Paper consists of THREE sections. 2. There is one question
More informationPRACTICE TEST ANSWER KEY & SCORING GUIDELINES GEOMETRY
Ohio s State Tests PRACTICE TEST ANSWER KEY & SCORING GUIDELINES GEOMETRY Table of Contents Questions 1 30: Content Summary and Answer Key... iii Question 1: Question and Scoring Guidelines... 1 Question
More informationGeometry Vocabulary Math Fundamentals Reference Sheet Page 1
Math Fundamentals Reference Sheet Page 1 Acute Angle An angle whose measure is between 0 and 90 Acute Triangle A that has all acute Adjacent Alternate Interior Angle Two coplanar with a common vertex and
More informationI can position figures in the coordinate plane for use in coordinate proofs. I can prove geometric concepts by using coordinate proof.
Page 1 of 14 Attendance Problems. 1. Find the midpoint between (0, x) and (y, z).. One leg of a right triangle has length 1, and the hypotenuse has length 13. What is the length of the other leg? 3. Find
More informationPerimeter and Area of Inscribed and Circumscribed Polygons
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-007 Perimeter and Area of Inscribed and Circumscribed
More informationCalculate the area of each figure. Each square on the grid represents a square that is one meter long and one meter wide.
CH 3 Test Review Boundary Lines: Area of Parallelograms and Triangles Calculate the area of each figure Each square on the grid represents a square that is one meter long and one meter wide 1 You are making
More informationGeometry Learning Targets
Geometry Learning Targets 2015 2016 G0. Algebra Prior Knowledge G0a. Simplify algebraic expressions. G0b. Solve a multi-step equation. G0c. Graph a linear equation or find the equation of a line. G0d.
More informationPerimeter, Area, Surface Area, & Volume
Additional Options: Hide Multiple Choice Answers (Written Response) Open in Microsoft Word (add page breaks and/or edit questions) Generation Date: 11/25/2009 Generated By: Margaret Buell Copyright 2009
More informationCIRCLES ON TAKS NAME CLASS PD DUE
CIRCLES ON TAKS NAME CLASS PD DUE 1. On the calculator: Let s say the radius is 2. Find the area. Now let s double the radius to 4 and find the area. How do these two numbers relate? 2. The formula for
More informationGeometry: Semester 2 Practice Final Unofficial Worked Out Solutions by Earl Whitney
Geometry: Semester 2 Practice Final Unofficial Worked Out Solutions by Earl Whitney 1. Wrapping a string around a trash can measures the circumference of the trash can. Assuming the trash can is circular,
More informationThe Research- Driven Solution to Raise the Quality of High School Core Courses. Geometry. Course Outline
The Research- Driven Solution to Raise the Quality of High School Core Courses Course Outline Course Outline Page 2 of 5 0 1 2 3 4 5 ACT Course Standards A. Prerequisites 1. Skills Acquired by Students
More informationSection 4.1 Investigating Circles
Section 4.1 Investigating Circles A circle is formed when all the points in a plane that are the same distance away from a center point. The distance from the center of the circle to any point on the edge
More informationPark Forest Math Team. Meet #3. Self-study Packet
Park Forest Math Team Meet #3 Self-study Packet Problem Categories for this Meet (in addition to topics of earlier meets): 1. Mystery: Problem solving 2. : Properties of Polygons, Pythagorean Theorem 3.
More informationUnit Lesson Plan: Measuring Length and Area: Area of shapes
Unit Lesson Plan: Measuring Length and Area: Area of shapes Day 1: Area of Square, Rectangles, and Parallelograms Day 2: Area of Triangles Trapezoids, Rhombuses, and Kites Day 3: Quiz over Area of those
More informationGeometry Geometry Grade Grade Grade
Grade Grade Grade 6.G.1 Find the area of right triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the
More informationMake geometric constructions. (Formalize and explain processes)
Standard 5: Geometry Pre-Algebra Plus Algebra Geometry Algebra II Fourth Course Benchmark 1 - Benchmark 1 - Benchmark 1 - Part 3 Draw construct, and describe geometrical figures and describe the relationships
More informationAny questions about the material so far? About the exercises?
Any questions about the material so far? About the exercises? Here is a question for you. In the diagram on the board, DE is parallel to AC, DB = 4, AB = 9 and BE = 8. What is the length EC? Polygons Definitions:
More informationVocabulary: Looking For Pythagoras
Vocabulary: Looking For Pythagoras Concept Finding areas of squares and other figures by subdividing or enclosing: These strategies for finding areas were developed in Covering and Surrounding. Students
More informationNeed more help with decimal subtraction? See T23. Note: The inequality sign is reversed only when multiplying or dividing by a negative number.
. (D) According to the histogram, junior boys sleep an average of.5 hours on a daily basis and junior girls sleep an average of. hours. To find how many more hours the average junior boy sleeps than the
More informationSolution Guide for Chapter 20
Solution Guide for Chapter 0 Here are the solutions for the Doing the Math exercises in Girls Get Curves! DTM from p. 351-35. In the diagram SLICE, LC and IE are altitudes of the triangle!sci. L I If SI
More informationMoore Catholic High School Math Department
Moore Catholic High School Math Department Geometry Vocabulary The following is a list of terms and properties which are necessary for success in a Geometry class. You will be tested on these terms during
More information8.4 Special Right Triangles
8.4. Special Right Triangles www.ck1.org 8.4 Special Right Triangles Learning Objectives Identify and use the ratios involved with isosceles right triangles. Identify and use the ratios involved with 30-60-90
More informationMath-2 Lesson 8-7: Unit 5 Review (Part -2)
Math- Lesson 8-7: Unit 5 Review (Part -) Trigonometric Functions sin cos A A SOH-CAH-TOA Some old horse caught another horse taking oats away. opposite ( length ) o sin A hypotenuse ( length ) h SOH adjacent
More informationFormative Assessment Area Unit 5
Area Unit 5 Administer the formative assessment and select contrasting student responses to create further opportunities for learning about area measure, especially the difference between units of length
More informationCongruent triangles/polygons : All pairs of corresponding parts are congruent; if two figures have the same size and shape.
Jan Lui Adv Geometry Ch 3: Congruent Triangles 3.1 What Are Congruent Figures? Congruent triangles/polygons : All pairs of corresponding parts are congruent; if two figures have the same size and shape.
More informationGeometry Curriculum Guide Lunenburg County Public Schools June 2014
Marking Period: 1 Days: 4 Reporting Category/Strand: Reasoning, Lines, and Transformations SOL G.1 The student will construct and judge the validity of a logical argument consisting of a set of premises
More informationMath 2 Plane Geometry part 1 Unit Updated January 13, 2017
Complementary angles (two angles whose sum is 90 ) and supplementary angles (two angles whose sum is 180. A straight line = 180. In the figure below and to the left, angle EFH and angle HFG form a straight
More informationTheta Circles & Polygons 2015 Answer Key 11. C 2. E 13. D 4. B 15. B 6. C 17. A 18. A 9. D 10. D 12. C 14. A 16. D
Theta Circles & Polygons 2015 Answer Key 1. C 2. E 3. D 4. B 5. B 6. C 7. A 8. A 9. D 10. D 11. C 12. C 13. D 14. A 15. B 16. D 17. A 18. A 19. A 20. B 21. B 22. C 23. A 24. C 25. C 26. A 27. C 28. A 29.
More informationGeometry I Can Statements I can describe the undefined terms: point, line, and distance along a line in a plane I can describe the undefined terms:
Geometry I Can Statements I can describe the undefined terms: point, line, and distance along a line in a plane I can describe the undefined terms: point, line, and distance along a line in a plane I can
More informationGeometry. Cluster: Experiment with transformations in the plane. G.CO.1 G.CO.2. Common Core Institute
Geometry Cluster: Experiment with transformations in the plane. G.CO.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of
More information7-3 Parallel and Perpendicular Lines
7-3 Parallel and Perpendicular Lines Interior Angles: Exterior Angles: Corresponding Angles: Vertical Angles: Supplementary Angles: 1 2 3 4 5 6 7 8 Complimentary Angles: 7-3 Parallel and Perpendicular
More informationSurface Area and Volume
Surface Area and Volume Day 1 - Surface Area of Prisms Surface Area = The total area of the surface of a three-dimensional object (Or think of it as the amount of paper you ll need to wrap the shape.)
More informationHoly Family Catholic High School. Geometry Review
Holy Family Catholic High School Geometry Review Version 0.1 Last Modified /17/008 THERE WILL BE NO FORMULAS GIVEN TO YOU DURING THE TEST! This will be a review of some of the topics needed on the Holy
More informationCommon Core Specifications for Geometry
1 Common Core Specifications for Geometry Examples of how to read the red references: Congruence (G-Co) 2-03 indicates this spec is implemented in Unit 3, Lesson 2. IDT_C indicates that this spec is implemented
More informationJanuary Regional Geometry Team: Question #1. January Regional Geometry Team: Question #2
January Regional Geometry Team: Question #1 Points P, Q, R, S, and T lie in the plane with S on and R on. If PQ = 5, PS = 3, PR = 5, QS = 3, and RT = 4, what is ST? 3 January Regional Geometry Team: Question
More informationPre-Algebra, Unit 10: Measurement, Area, and Volume Notes
Pre-Algebra, Unit 0: Measurement, Area, and Volume Notes Triangles, Quadrilaterals, and Polygons Objective: (4.6) The student will classify polygons. Take this opportunity to review vocabulary and previous
More informationGeometry Vocabulary. Name Class
Geometry Vocabulary Name Class Definition/Description Symbol/Sketch 1 point An exact location in space. In two dimensions, an ordered pair specifies a point in a coordinate plane: (x,y) 2 line 3a line
More informationPOLYGONS
POLYGONS 8.1.1 8.1.5 After studying triangles and quadrilaterals, the students now extend their knowledge to all polygons. A polygon is a closed, two-dimensional figure made of three or more non-intersecting
More information2 Formula (given): Volume of a Pyramid V = 1/3 BH What does B represent? Formula: Area of a Trapezoid. 3 Centroid. 4 Midsegment of a triangle
1 Formula: Area of a Trapezoid 2 Formula (given): Volume of a Pyramid V = 1/3 BH What does B represent? 3 Centroid 4 Midsegment of a triangle 5 Slope formula 6 Point Slope Form of Linear Equation *can
More information2015 Theta Geometry Topic Test Answer Key 13. A 12. D 23. C 24. D 15. A 26. B 17. B 8. A 29. B 10. C 11. D 14. B 16. A
2015 Theta Geometry Topic Test Answer Key 1. A 2. D 3. C 4. D 5. A 6. B 7. B 8. A 9. B 10. C 11. D 12. D 13. A 14. B 15. A 16. A 17. B 18. E (9) 19. A 20. D 21. A 22. C 23. C 24. D 25. C 26. B 27. C 28.
More informationUnit 1: Fundamentals of Geometry
Name: 1 2 Unit 1: Fundamentals of Geometry Vocabulary Slope: m y x 2 2 Formulas- MUST KNOW THESE! y x 1 1 *Used to determine if lines are PARALLEL, PERPENDICULAR, OR NEITHER! Parallel Lines: SAME slopes
More information11.3 Surface Area of Pyramids and Cones
11.3 Surface Area of Pyramids and Cones Learning Objectives Find the surface area of a pyramid. Find the surface area of a cone. Review Queue 1. A rectangular prism has sides of 5 cm, 6 cm, and 7 cm. What
More information3. The sides of a rectangle are in ratio fo 3:5 and the rectangle s area is 135m2. Find the dimensions of the rectangle.
Geometry B Honors Chapter Practice Test 1. Find the area of a square whose diagonal is. 7. Find the area of the triangle. 60 o 12 2. Each rectangle garden below has an area of 0. 8. Find the area of the
More informationExample Items. Geometry
Example Items Geometry Geometry Example Items are a representative set of items for the ACP. Teachers may use this set of items along with the test blueprint as guides to prepare students for the ACP.
More informationPerimeter and Area. Slide 1 / 183. Slide 2 / 183. Slide 3 / 183. Table of Contents. New Jersey Center for Teaching and Learning
New Jersey Center for Teaching and Learning Slide 1 / 183 Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students
More informationSection Congruence Through Constructions
Section 10.1 - Congruence Through Constructions Definitions: Similar ( ) objects have the same shape but not necessarily the same size. Congruent ( =) objects have the same size as well as the same shape.
More informationA. 180 B. 108 C. 360 D. 540
Part I - Multiple Choice - Circle your answer: REVIEW FOR FINAL EXAM - GEOMETRY 2 1. Find the area of the shaded sector. Q O 8 P A. 2 π B. 4 π C. 8 π D. 16 π 2. An octagon has sides. A. five B. six C.
More informationUse of Number Maths Statement Code no: 1 Student: Class: At Junior Certificate level the student can: Apply the knowledge and skills necessary to perf
Use of Number Statement Code no: 1 Apply the knowledge and skills necessary to perform mathematical calculations 1 Recognise simple fractions, for example 1 /4, 1 /2, 3 /4 shown in picture or numerical
More informationSection 14: Trigonometry Part 1
Section 14: Trigonometry Part 1 The following Mathematics Florida Standards will be covered in this section: MAFS.912.F-TF.1.1 MAFS.912.F-TF.1.2 MAFS.912.F-TF.1.3 Understand radian measure of an angle
More informationMATHEMATICS Geometry Standard: Number, Number Sense and Operations
Standard: Number, Number Sense and Operations Number and Number A. Connect physical, verbal and symbolic representations of 1. Connect physical, verbal and symbolic representations of Systems integers,
More informationModule 1 Session 1 HS. Critical Areas for Traditional Geometry Page 1 of 6
Critical Areas for Traditional Geometry Page 1 of 6 There are six critical areas (units) for Traditional Geometry: Critical Area 1: Congruence, Proof, and Constructions In previous grades, students were
More informationMATHia Unit MATHia Workspace Overview TEKS
1 Tools of Geometry Lines, Rays, Segments, and Angles Distances on the Coordinate Plane Parallel and Perpendicular Lines Angle Properties Naming Lines, Rays, Segments, and Angles Working with Measures
More information