# Student Outcomes. Lesson Notes. Classwork. Exercises 1 3 (4 minutes)

Size: px
Start display at page:

## Transcription

1 Student Outcomes Students give an informal derivation of the relationship between the circumference and area of a circle. Students know the formula for the area of a circle and use it to solve problems. Lesson Notes Remind students of the definitions for circle and circumference from the previous lesson. The Opening Exercise is a lead-in to the derivation of the formula for the area of a circle. Not only do students need to know and be able to apply the formula for the area of a circle, it is critical for them to also be able to draw the diagram associated with each problem in order to solve it successfully. Students must be able to translate words into mathematical expressions and equations and be able to determine which parts of the problem are known and which are unknown or missing. Classwork Exercises 1 3 (4 minutes) Exercises 1 3 Solve the problem below individually. Explain your solution. 1. Find the radius of a circle if its circumference is inches. Use π If C = 2πr, then = 2πr. Solving the equation for r, The radius of the circle is approximately 6 in = 2πr ( 1 1 ) = ( 2π 2π ) 2πr 1 (37. 68) r r 2. Determine the area of the rectangle below. Name two ways that can be used to find the area of the rectangle. The area of the rectangle is 24 cm 2. The area can be found by counting the square units inside the rectangle or by multiplying the length (6 cm) by the width (4 cm). 252

2 3. Find the length of a rectangle if the area is 27 cm 2 and the width is 3 cm. If the area of the rectangle is Area = length width, then 27 cm 2 = l 3 cm cm2 = 1 l 3 cm 3 9 cm = l Exploratory Challenge (10 minutes) Complete the exercise below. Exploratory Challenge To find the formula for the area of a circle, cut a circle into 16 equal pieces. Scaffolding: Provide a circle divided into 16 equal sections for students to cut out and reassemble as a rectangle. MP.7 Arrange the triangular wedges by alternating the triangle directions and sliding them together to make a parallelogram. Cut the triangle on the left side in half on the given line, and slide the outside half of the triangle to the other end of the parallelogram in order to create an approximate rectangle. Move half to the other end. The circumference is 2πr, where the radius is r. Therefore, half of the circumference is πr. 253

3 What is the area of the rectangle using the side lengths above? The area of the rectangle is base times height, and, in this case, A = πr r. Are the areas of the rectangle and the circle the same? Yes, since we just rearranged pieces of the circle to make the rectangle, the area of the rectangle and the area of the circle are approximately equal. Note that the more sections we cut the circle into, the closer the approximation. If the area of the rectangular shape and the circle are the same, what is the area of the circle? The area of a circle is written as A = πr r, or A = πr 2. Example 1 (4 minutes) Example 1 Use the shaded square centimeter units to approximate the area of the circle. What is the radius of the circle? 10 cm What would be a quicker method for determining the area of the circle other than counting all of the squares in the entire circle? Count 1 of the squares needed; then, multiply that by four in order to determine the area of the entire circle. 4 Using the diagram, how many squares were used to cover one-fourth of the circle? The area of one-fourth of the circle is approximately 79 cm 2. What is the area of the entire circle? A 4 79 cm 2 A 316 cm 2 254

4 Example 2 (4 minutes) Example 2 A sprinkler rotates in a circular pattern and sprays water over a distance of 12 feet. What is the area of the circular region covered by the sprinkler? Express your answer to the nearest square foot. Draw a diagram to assist you in solving the problem. What does the distance of 12 feet represent in this problem? The radius is 12 feet. What information is needed to solve the problem? The formula to find the area of a circle is A = πr 2. If the radius is 12 ft., then A = π (12 ft. ) 2 = 144π ft 2, or approximately 452 ft 2. Make a point of telling students that an answer in exact form is in terms of π, not substituting an approximation of pi. Example 3 (4 minutes) Example 3 Suzanne is making a circular table out of a square piece of wood. The radius of the circle that she is cutting is 3 feet. How much waste will she have for this project? Express your answer to the nearest square foot. Draw a diagram to assist you in solving the problem. What does the distance of 3 feet represent in this problem? The radius of the circle is 3 feet. What information is needed to solve the problem? The area of the circle and the area of the square are needed so that we can subtract the area of the circle from the area of the square to determine the amount of waste. What information do we need to determine the area of the square and the circle? Circle: just radius because A = πr 2 Square: one side length 255

5 How will we determine the waste? The waste is the area left over from the square after cutting out the circular region. The area of the circle is A = π (3 ft. ) 2 = 9π ft ft 2. The area of the square is found by first finding the diameter of the circle, which is the same as the side of the square. The diameter is d = 2r; so, d = 2 3 ft. or 6 ft. The area of a square is found by multiplying the length and width, so A = 6 ft. 6 ft. = 36 ft 2. The solution is the difference between the area of the square and the area of the circle, so 36 ft ft ft 2. Does your solution answer the problem as stated? Yes, the amount of waste is ft 2. Exercises 4 6 (11 minutes) Solve in cooperative groups of two or three. Exercises A circle has a radius of 2 cm. a. Find the exact area of the circular region. A = π (2 cm) 2 = 4π cm 2 b. Find the approximate area using to approximate π. A = 4 cm 2 π 4 cm cm 2 5. A circle has a radius of 7 cm. a. Find the exact area of the circular region. A = π (7 cm) 2 = 49π cm 2 b. Find the approximate area using 22 to approximate π. 7 A = 49 π cm 2 ( ) cm2 154 cm 2 c. What is the circumference of the circle? C = 2π 7 cm = 14π cm cm 6. Joan determined that the area of the circle below is 400π cm 2. Melinda says that Joan s solution is incorrect; she believes that the area is 100π cm 2. Who is correct and why? Melinda is correct. Joan found the area by multiplying π by the square of 20 cm (which is the diameter) to get a result of 400π cm 2, which is incorrect. Melinda found that the radius was 10 cm (half of the diameter). Melinda multiplied π by the square of the radius to get a result of 100π cm

6 Closing (4 minutes) Strategies for problem solving include drawing a diagram to represent the problem and identifying the given information and needed information to solve the problem. Using the original circle in this lesson, cut it into 64 equal slices. Reassemble the figure. What do you notice? It looks more like a rectangle. Ask students to imagine repeating the slicing into even thinner slices (infinitely thin). Then, ask the next two questions. What does the length of the rectangle become? An approximation of half of the circumference of the circle What does the width of the rectangle become? An approximation of the radius Thus, we conclude that the area of the circle is A = 1 2 Cr. If A = 1 2 Cr, then A = 1 2 2πr r or A = πr2. Also see video link: Relevant Vocabulary CIRCULAR REGION (OR DISK): Given a point C in the plane and a number r > 0, the circular region (or disk) with center C and radius r is the set of all points in the plane whose distance from the point C is less than or equal to r. The boundary of a disk is a circle. The area of a circle refers to the area of the disk defined by the circle. Exit Ticket (4 minutes) 257

7 Name Date Exit Ticket Complete each statement using the words or algebraic expressions listed in the word bank below. 1. The length of the of the rectangular region approximates the length of the of the circle. 2. The of the rectangle approximates the length of one-half of the circumference of the circle. 3. The circumference of the circle is. 4. The of the is 2r. 5. The ratio of the circumference to the diameter is. 6. Area (circle) = Area of ( ) = 1 2 circumference r = 1 (2πr) r = π r r =. 2 Word bank radius height base 2πr diameter circle rectangle πr 2 π 258

8 Exit Ticket Sample Solutions Complete each statement using the words or algebraic expressions listed in the word bank below. 1. The length of the height of the rectangular region approximates the length of the radius of the circle. 2. The base of the rectangle approximates the length of one-half of the circumference of the circle. 3. The circumference of the circle is 2πr. 4. The diameter of the circle is 2r. 5. The ratio of the circumference to the diameter is π. 6. Area (circle) = Area of (rectangle) = 1 2 circumference r = 1 2 (2πr) r = π r r = πr2. Problem Set Sample Solutions 1. The following circles are not drawn to scale. Find the area of each circle. (Use 22 as an approximation for π.) cm cm 2 5, ft 2 1, cm 2 2. A circle has a diameter of 20 inches. a. Find the exact area, and find an approximate area using π If the diameter is 20 in., then the radius is 10 in. If A = πr 2, then A = π (10 in. ) 2 or 100π in 2. A ( ) in in 2. b. What is the circumference of the circle using π 3. 14? If the diameter is 20 in., then the circumference is C = πd or C in in. 3. A circle has a diameter of 11 inches. a. Find the exact area and an approximate area using π If the diameter is 11 in., then the radius is 11 2 in. If A = πr2, then A = π ( 11 2 A ( ) in in 2 in. )2 or π in2. b. What is the circumference of the circle using π 3. 14? If the diameter is 11 inches, then the circumference is C = πd or C in in. 259

9 4. Using the figure below, find the area of the circle. In this circle, the diameter is the same as the length of the side of the square. The diameter is 10 cm; so, the radius is 5 cm. A = πr 2, so A = π(5 cm) 2 = 25π cm A path bounds a circular lawn at a park. If the inner edge of the path is 132 ft. around, approximate the amount of area of the lawn inside the circular path. Use π The length of the path is the same as the circumference. Find the radius from the circumference; then, find the area. C = 2πr 132 ft r 132 ft r ft r 21 ft. r A 22 7 (21 ft. )2 A 1386 ft 2 6. The area of a circle is 36π cm 2. Find its circumference. Find the radius from the area of the circle; then, use it to find the circumference. A = πr 2 36π cm 2 = πr 2 1 π 36π cm2 = 1 π πr2 36 cm 2 = r 2 6 cm = r C = 2πr C = 2π 6 cm C = 12π cm 7. Find the ratio of the area of two circles with radii 3 cm and 4 cm. The area of the circle with radius 3 cm is 9π cm 2. The area of the circle with the radius 4 cm is 16π cm 2. The ratio of the area of the two circles is 9π 16π or

10 8. If one circle has a diameter of 10 cm and a second circle has a diameter of 20 cm, what is the ratio of the area of the larger circle to the area of the smaller circle? The area of the circle with the diameter of 10 cm has a radius of 5 cm. The area of the circle with the diameter of 10 cm is π (5 cm) 2, or 25π cm 2. The area of the circle with the diameter of 20 cm has a radius of 10 cm. The area of the circle with the diameter of 20 cm is π (10 cm) 2 or 100π cm 2. The ratio of the diameters is 20 to 10 or 2: 1, while the ratio of the areas is 100π to 25π or 4: Describe a rectangle whose perimeter is 132 ft. and whose area is less than 1 ft 2. Is it possible to find a circle whose circumference is 132 ft. and whose area is less than 1 ft 2? If not, provide an example or write a sentence explaining why no such circle exists. A rectangle that has a perimeter of 132 ft. can have a length of ft. and a width of ft. The area of such a rectangle is ft 2, which is less than 1 ft 2. No, because a circle that has a circumference of 132 ft. has a radius of approximately 21 ft. A = πr 2 = π(21) 2 = If the diameter of a circle is double the diameter of a second circle, what is the ratio of the area of the first circle to the area of the second? If I choose a diameter of 24 cm for the first circle, then the diameter of the second circle is 12 cm. The first circle has a radius of 12 cm and an area of 144π cm 2. The second circle has a radius of 6 cm and an area of 36π cm 2. The ratio of the area of the first circle to the second is 144π to 36π, which is a 4 to 1 ratio. The ratio of the diameters is 2, while the ratio of the areas is the square of 2, or

### Characteristics of the Four Main Geometrical Figures

Math 40 9.7 & 9.8: The Big Four Square, Rectangle, Triangle, Circle Pre Algebra We will be focusing our attention on the formulas for the area and perimeter of a square, rectangle, triangle, and a circle.

### Show that when a circle is inscribed inside a square the diameter of the circle is the same length as the side of the square.

Week & Day Week 6 Day 1 Concept/Skill Perimeter of a square when given the radius of an inscribed circle Standard 7.MG:2.1 Use formulas routinely for finding the perimeter and area of basic twodimensional

### PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.

PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the

### Perimeter is the length of the boundary of a two dimensional figure.

Section 2.2: Perimeter and Area Perimeter is the length of the boundary of a two dimensional figure. The perimeter of a circle is called the circumference. The perimeter of any two dimensional figure whose

### Arc Length and Areas of Sectors

Student Outcomes When students are provided with the angle measure of the arc and the length of the radius of the circle, they understand how to determine the length of an arc and the area of a sector.

### Calculating Area, Perimeter and Volume

Calculating Area, Perimeter and Volume You will be given a formula table to complete your math assessment; however, we strongly recommend that you memorize the following formulae which will be used regularly

### GAP CLOSING. 2D Measurement. Intermediate / Senior Student Book

GAP CLOSING 2D Measurement Intermediate / Senior Student Book 2-D Measurement Diagnostic...3 Areas of Parallelograms, Triangles, and Trapezoids...6 Areas of Composite Shapes...14 Circumferences and Areas

### Geometry Unit 6 Areas and Perimeters

Geometry Unit 6 Areas and Perimeters Name Lesson 8.1: Areas of Rectangle (and Square) and Parallelograms How do we measure areas? Area is measured in square units. The type of the square unit you choose

### Perimeter, Area, and Volume

Perimeter, Area, and Volume Perimeter of Common Geometric Figures The perimeter of a geometric figure is defined as the distance around the outside of the figure. Perimeter is calculated by adding all

### Tallahassee Community College PERIMETER

Tallahassee Community College 47 PERIMETER The perimeter of a plane figure is the distance around it. Perimeter is measured in linear units because we are finding the total of the lengths of the sides

### 43 Perimeter and Area

43 Perimeter and Area Perimeters of figures are encountered in real life situations. For example, one might want to know what length of fence will enclose a rectangular field. In this section we will study

### Postulate 17 The area of a square is the square of the length of a. Postulate 18 If two figures are congruent, then they have the same.

Chapter 11: Areas of Plane Figures (page 422) 11-1: Areas of Rectangles (page 423) Rectangle Rectangular Region Area is measured in units. Postulate 17 The area of a square is the square of the length

### Geometry Notes VOLUME AND SURFACE AREA

Volume and Surface Area Page 1 of 19 VOLUME AND SURFACE AREA Objectives: After completing this section, you should be able to do the following: Calculate the volume of given geometric figures. Calculate

### Section 7.2 Area. The Area of Rectangles and Triangles

Section 7. Area The Area of Rectangles and Triangles We encounter two dimensional objects all the time. We see objects that take on the shapes similar to squares, rectangle, trapezoids, triangles, and

### Area and Circumference

4.4 Area and Circumference 4.4 OBJECTIVES 1. Use p to find the circumference of a circle 2. Use p to find the area of a circle 3. Find the area of a parallelogram 4. Find the area of a triangle 5. Convert

### Circumference and Area of a Circle

Overview Math Concepts Materials Students explore how to derive pi (π) as a ratio. Students also study the circumference and area of a circle using formulas. numbers and operations TI-30XS MultiView two-dimensional

### Circumference of a Circle

Circumference of a Circle A circle is a shape with all points the same distance from the center. It is named by the center. The circle to the left is called circle A since the center is at point A. If

### MATH STUDENT BOOK. 6th Grade Unit 8

MATH STUDENT BOOK 6th Grade Unit 8 Unit 8 Geometry and Measurement MATH 608 Geometry and Measurement INTRODUCTION 3 1. PLANE FIGURES 5 PERIMETER 5 AREA OF PARALLELOGRAMS 11 AREA OF TRIANGLES 17 AREA OF

### Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

### Cylinder Volume Lesson Plan

Cylinder Volume Lesson Plan Concept/principle to be demonstrated: This lesson will demonstrate the relationship between the diameter of a circle and its circumference, and impact on area. The simplest

### Geometry and Measurement

The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

### Free Pre-Algebra Lesson 55! page 1

Free Pre-Algebra Lesson 55! page 1 Lesson 55 Perimeter Problems with Related Variables Take your skill at word problems to a new level in this section. All the problems are the same type, so that you can

### Kristen Kachurek. Circumference, Perimeter, and Area Grades 7-10 5 Day lesson plan. Technology and Manipulatives used:

Kristen Kachurek Circumference, Perimeter, and Area Grades 7-10 5 Day lesson plan Technology and Manipulatives used: TI-83 Plus calculator Area Form application (for TI-83 Plus calculator) Login application

### Geometry - Calculating Area and Perimeter

Geometry - Calculating Area and Perimeter In order to complete any of mechanical trades assessments, you will need to memorize certain formulas. These are listed below: (The formulas for circle geometry

### Circumference CHAPTER. www.ck12.org 1

www.ck12.org 1 CHAPTER 1 Circumference Here you ll learn how to find the distance around, or the circumference of, a circle. What if you were given the radius or diameter of a circle? How could you find

### Perimeter. 14ft. 5ft. 11ft.

Perimeter The perimeter of a geometric figure is the distance around the figure. The perimeter could be thought of as walking around the figure while keeping track of the distance traveled. To determine

### B = 1 14 12 = 84 in2. Since h = 20 in then the total volume is. V = 84 20 = 1680 in 3

45 Volume Surface area measures the area of the two-dimensional boundary of a threedimensional figure; it is the area of the outside surface of a solid. Volume, on the other hand, is a measure of the space

### Inv 1 5. Draw 2 different shapes, each with an area of 15 square units and perimeter of 16 units.

Covering and Surrounding: Homework Examples from ACE Investigation 1: Questions 5, 8, 21 Investigation 2: Questions 6, 7, 11, 27 Investigation 3: Questions 6, 8, 11 Investigation 5: Questions 15, 26 ACE

### Calculating the Surface Area of a Cylinder

Calculating the Measurement Calculating The Surface Area of a Cylinder PRESENTED BY CANADA GOOSE Mathematics, Grade 8 Introduction Welcome to today s topic Parts of Presentation, questions, Q&A Housekeeping

### Teacher Page Key. Geometry / Day # 13 Composite Figures 45 Min.

Teacher Page Key Geometry / Day # 13 Composite Figures 45 Min. 9-1.G.1. Find the area and perimeter of a geometric figure composed of a combination of two or more rectangles, triangles, and/or semicircles

### Area of Parallelograms (pages 546 549)

A Area of Parallelograms (pages 546 549) A parallelogram is a quadrilateral with two pairs of parallel sides. The base is any one of the sides and the height is the shortest distance (the length of a perpendicular

### Finding Volume of Rectangular Prisms

MA.FL.7.G.2.1 Justify and apply formulas for surface area and volume of pyramids, prisms, cylinders, and cones. MA.7.G.2.2 Use formulas to find surface areas and volume of three-dimensional composite shapes.

### 7.4A/7.4B STUDENT ACTIVITY #1

7.4A/7.4B STUDENT ACTIVITY #1 Write a formula that could be used to find the radius of a circle, r, given the circumference of the circle, C. The formula in the Grade 7 Mathematics Chart that relates the

### Area of Parallelograms, Triangles, and Trapezoids (pages 314 318)

Area of Parallelograms, Triangles, and Trapezoids (pages 34 38) Any side of a parallelogram or triangle can be used as a base. The altitude of a parallelogram is a line segment perpendicular to the base

### Algebra Geometry Glossary. 90 angle

lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:

### Lesson 13: The Formulas for Volume

Student Outcomes Students develop, understand, and apply formulas for finding the volume of right rectangular prisms and cubes. Lesson Notes This lesson is a continuation of Lessons 11, 12, and Module

### GAP CLOSING. 2D Measurement GAP CLOSING. Intermeditate / Senior Facilitator s Guide. 2D Measurement

GAP CLOSING 2D Measurement GAP CLOSING 2D Measurement Intermeditate / Senior Facilitator s Guide 2-D Measurement Diagnostic...4 Administer the diagnostic...4 Using diagnostic results to personalize interventions...4

### Finding Areas of Shapes

Baking Math Learning Centre Finding Areas of Shapes Bakers often need to know the area of a shape in order to plan their work. A few formulas are required to find area. First, some vocabulary: Diameter

### How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

### Unit 7 Circles. Vocabulary and Formulas for Circles:

ccelerated G Unit 7 ircles Name & ate Vocabulary and Formulas for ircles: irections: onsider 1) Find the circumference of the circle. to answer the following questions. Exact: pproximate: 2) Find the area

### MATH 110 Landscape Horticulture Worksheet #4

MATH 110 Landscape Horticulture Worksheet #4 Ratios The math name for a fraction is ratio. It is just a comparison of one quantity with another quantity that is similar. As a Landscape Horticulturist,

### GAP CLOSING. Volume and Surface Area. Intermediate / Senior Student Book

GAP CLOSING Volume and Surface Area Intermediate / Senior Student Book Volume and Surface Area Diagnostic...3 Volumes of Prisms...6 Volumes of Cylinders...13 Surface Areas of Prisms and Cylinders...18

### Lesson 21. Circles. Objectives

Student Name: Date: Contact Person Name: Phone Number: Lesson 1 Circles Objectives Understand the concepts of radius and diameter Determine the circumference of a circle, given the diameter or radius Determine

### 9 Areas and Perimeters

9 Areas and Perimeters This is is our next key Geometry unit. In it we will recap some of the concepts we have met before. We will also begin to develop a more algebraic approach to finding areas and perimeters.

### 9 Area, Perimeter and Volume

9 Area, Perimeter and Volume 9.1 2-D Shapes The following table gives the names of some 2-D shapes. In this section we will consider the properties of some of these shapes. Rectangle All angles are right

### 12 Surface Area and Volume

12 Surface Area and Volume 12.1 Three-Dimensional Figures 12.2 Surface Areas of Prisms and Cylinders 12.3 Surface Areas of Pyramids and Cones 12.4 Volumes of Prisms and Cylinders 12.5 Volumes of Pyramids

### Volume of Right Prisms Objective To provide experiences with using a formula for the volume of right prisms.

Volume of Right Prisms Objective To provide experiences with using a formula for the volume of right prisms. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game

### Area of a triangle: The area of a triangle can be found with the following formula: 1. 2. 3. 12in

Area Review Area of a triangle: The area of a triangle can be found with the following formula: 1 A 2 bh or A bh 2 Solve: Find the area of each triangle. 1. 2. 3. 5in4in 11in 12in 9in 21in 14in 19in 13in

### Area is a measure of how much space is occupied by a figure. 1cm 1cm

Area Area is a measure of how much space is occupied by a figure. Area is measured in square units. For example, one square centimeter (cm ) is 1cm wide and 1cm tall. 1cm 1cm A figure s area is the number

### Area of a triangle: The area of a triangle can be found with the following formula: You can see why this works with the following diagrams:

Area Review Area of a triangle: The area of a triangle can be found with the following formula: 1 A 2 bh or A bh 2 You can see why this works with the following diagrams: h h b b Solve: Find the area of

### SURFACE AREA AND VOLUME

SURFACE AREA AND VOLUME In this unit, we will learn to find the surface area and volume of the following threedimensional solids:. Prisms. Pyramids 3. Cylinders 4. Cones It is assumed that the reader has

### Session 7 Circles and Pi (π)

Key Terms in This Session Session 7 Circles and Pi (π) Previously Introduced accuracy area precision scale factor similar figures New in This Session circumference diameter irrational number perimeter

### Circumference Pi Regular polygon. Dates, assignments, and quizzes subject to change without advance notice.

Name: Period GPreAP UNIT 14: PERIMETER AND AREA I can define, identify and illustrate the following terms: Perimeter Area Base Height Diameter Radius Circumference Pi Regular polygon Apothem Composite

### Chapter 7 Quiz. (1.) Which type of unit can be used to measure the area of a region centimeter, square centimeter, or cubic centimeter?

Chapter Quiz Section.1 Area and Initial Postulates (1.) Which type of unit can be used to measure the area of a region centimeter, square centimeter, or cubic centimeter? (.) TRUE or FALSE: If two plane

### 16 Circles and Cylinders

16 Circles and Cylinders 16.1 Introduction to Circles In this section we consider the circle, looking at drawing circles and at the lines that split circles into different parts. A chord joins any two

### Solids. Objective A: Volume of a Solids

Solids Math00 Objective A: Volume of a Solids Geometric solids are figures in space. Five common geometric solids are the rectangular solid, the sphere, the cylinder, the cone and the pyramid. A rectangular

### 1. Kyle stacks 30 sheets of paper as shown to the right. Each sheet weighs about 5 g. How can you find the weight of the whole stack?

Prisms and Cylinders Answer Key Vocabulary: cylinder, height (of a cylinder or prism), prism, volume Prior Knowledge Questions (Do these BEFORE using the Gizmo.) [Note: The purpose of these questions is

### YOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS WITHOUT A CALCULATOR!

DETAILED SOLUTIONS AND CONCEPTS - SIMPLE GEOMETRIC FIGURES Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! YOU MUST

### MENSURATION. Definition

MENSURATION Definition 1. Mensuration : It is a branch of mathematics which deals with the lengths of lines, areas of surfaces and volumes of solids. 2. Plane Mensuration : It deals with the sides, perimeters

8 th Grade Task 2 Rugs Student Task Core Idea 4 Geometry and Measurement Find perimeters of shapes. Use Pythagorean theorem to find side lengths. Apply appropriate techniques, tools and formulas to determine

### FCAT FLORIDA COMPREHENSIVE ASSESSMENT TEST. Mathematics Reference Sheets. Copyright Statement for this Assessment and Evaluation Services Publication

FCAT FLORIDA COMPREHENSIVE ASSESSMENT TEST Mathematics Reference Sheets Copyright Statement for this Assessment and Evaluation Services Publication Authorization for reproduction of this document is hereby

### 6-4 : Learn to find the area and circumference of circles. Area and Circumference of Circles (including word problems)

Circles 6-4 : Learn to fin the area an circumference of circles. Area an Circumference of Circles (incluing wor problems) 8-3 Learn to fin the Circumference of a circle. 8-6 Learn to fin the area of circles.

### SA B 1 p where is the slant height of the pyramid. V 1 3 Bh. 3D Solids Pyramids and Cones. Surface Area and Volume of a Pyramid

Accelerated AAG 3D Solids Pyramids and Cones Name & Date Surface Area and Volume of a Pyramid The surface area of a regular pyramid is given by the formula SA B 1 p where is the slant height of the pyramid.

### Developing Conceptual Understanding of Number. Set J: Perimeter and Area

Developing Conceptual Understanding of Number Set J: Perimeter and Area Carole Bilyk cbilyk@gov.mb.ca Wayne Watt wwatt@mts.net Perimeter and Area Vocabulary perimeter area centimetres right angle Notes

### 2. Complete the table to identify the effect tripling the radius of a cylinder s base has on its volume. Cylinder Height (cm) h

Name: Period: Date: K. Williams ID: A 8th Grade Chapter 14 TEST REVIEW 1. Determine the volume of the cylinder. Use 3.14 for. 2. Complete the table to identify the effect tripling the radius of a cylinder

### Pizza! Pizza! Assessment

Pizza! Pizza! Assessment 1. A local pizza restaurant sends pizzas to the high school twelve to a carton. If the pizzas are one inch thick, what is the volume of the cylindrical shipping carton for the

### The small increase in x is. and the corresponding increase in y is. Therefore

Differentials For a while now, we have been using the notation dy to mean the derivative of y with respect to. Here is any variable, and y is a variable whose value depends on. One of the reasons that

### Filling and Wrapping: Homework Examples from ACE

Filling and Wrapping: Homework Examples from ACE Investigation 1: Building Smart Boxes: Rectangular Prisms, ACE #3 Investigation 2: Polygonal Prisms, ACE #12 Investigation 3: Area and Circumference of

### CALCULATING THE AREA OF A FLOWER BED AND CALCULATING NUMBER OF PLANTS NEEDED

This resource has been produced as a result of a grant awarded by LSIS. The grant was made available through the Skills for Life Support Programme in 2010. The resource has been developed by (managers

### Grade 8 Mathematics Measurement: Lesson 6

Grade 8 Mathematics Measurement: Lesson 6 Read aloud to the students the material that is printed in boldface type inside the boxes. Information in regular type inside the boxes and all information outside

### All I Ever Wanted to Know About Circles

Parts of the Circle: All I Ever Wanted to Know About Circles 1. 2. 3. Important Circle Vocabulary: CIRCLE- the set off all points that are the distance from a given point called the CENTER- the given from

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Santa Monica College COMPASS Geometry Sample Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the area of the shaded region. 1) 5 yd 6 yd

### 12-1 Representations of Three-Dimensional Figures

Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 12-1 Representations of Three-Dimensional Figures Use isometric dot paper to sketch each prism. 1. triangular

### 2006 Geometry Form A Page 1

2006 Geometry Form Page 1 1. he hypotenuse of a right triangle is 12" long, and one of the acute angles measures 30 degrees. he length of the shorter leg must be: () 4 3 inches () 6 3 inches () 5 inches

### 7 Literal Equations and

CHAPTER 7 Literal Equations and Inequalities Chapter Outline 7.1 LITERAL EQUATIONS 7.2 INEQUALITIES 7.3 INEQUALITIES USING MULTIPLICATION AND DIVISION 7.4 MULTI-STEP INEQUALITIES 113 7.1. Literal Equations

### Geometry Solve real life and mathematical problems involving angle measure, area, surface area and volume.

Performance Assessment Task Pizza Crusts Grade 7 This task challenges a student to calculate area and perimeters of squares and rectangles and find circumference and area of a circle. Students must find

### Applications for Triangles

Not drawn to scale Applications for Triangles 1. 36 in. 40 in. 33 in. 1188 in. 2 69 in. 2 138 in. 2 1440 in. 2 2. 188 in. 2 278 in. 2 322 in. 2 none of these Find the area of a parallelogram with the given

### VOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region.

Math 6 NOTES 7.5 Name VOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region. **The formula for the volume of a rectangular prism is:** l = length w = width h = height Study Tip:

### WEIGHTS AND MEASURES. Linear Measure. 1 Foot12 inches. 1 Yard 3 feet - 36 inches. 1 Rod 5 1/2 yards - 16 1/2 feet

WEIGHTS AND MEASURES Linear Measure 1 Foot12 inches 1 Yard 3 feet - 36 inches 1 Rod 5 1/2 yards - 16 1/2 feet 1 Furlong 40 rods - 220 yards - 660 feet 1 Mile 8 furlongs - 320 rods - 1,760 yards 5,280 feet

### The teacher gives the student a ruler, shows her the shape below and asks the student to calculate the shape s area.

Complex area Georgia is able to calculate the area of a complex shape by mentally separating the shape into familiar shapes. She is able to use her knowledge of the formula for the area of a rectangle

### Area of Circles. Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required)

Area of Circles Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org

### Area. Area Overview. Define: Area:

Define: Area: Area Overview Kite: Parallelogram: Rectangle: Rhombus: Square: Trapezoid: Postulates/Theorems: Every closed region has an area. If closed figures are congruent, then their areas are equal.

### 2nd Semester Geometry Final Exam Review

Class: Date: 2nd Semester Geometry Final Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of an amusement park created a circular

### CHAPTER 8, GEOMETRY. 4. A circular cylinder has a circumference of 33 in. Use 22 as the approximate value of π and find the radius of this cylinder.

TEST A CHAPTER 8, GEOMETRY 1. A rectangular plot of ground is to be enclosed with 180 yd of fencing. If the plot is twice as long as it is wide, what are its dimensions? 2. A 4 cm by 6 cm rectangle has

### Chapter 19. Mensuration of Sphere

8 Chapter 19 19.1 Sphere: A sphere is a solid bounded by a closed surface every point of which is equidistant from a fixed point called the centre. Most familiar examples of a sphere are baseball, tennis

### Lesson 1: Introducing Circles

IRLES N VOLUME Lesson 1: Introducing ircles ommon ore Georgia Performance Standards M9 12.G..1 M9 12.G..2 Essential Questions 1. Why are all circles similar? 2. What are the relationships among inscribed

### Calculating Perimeter

Calculating Perimeter and Area Formulas are equations used to make specific calculations. Common formulas (equations) include: P = 2l + 2w perimeter of a rectangle A = l + w area of a square or rectangle

### Objective: To distinguish between degree and radian measure, and to solve problems using both.

CHAPTER 3 LESSON 1 Teacher s Guide Radian Measure AW 3.2 MP 4.1 Objective: To distinguish between degree and radian measure, and to solve problems using both. Prerequisites Define the following concepts.

### Area, Perimeter, Volume and Pythagorean Theorem Assessment

Area, Perimeter, Volume and Pythagorean Theorem Assessment Name: 1. Find the perimeter of a right triangle with legs measuring 10 inches and 24 inches a. 34 inches b. 60 inches c. 120 inches d. 240 inches

### Grade 7 & 8 Math Circles Circles, Circles, Circles March 19/20, 2013

Faculty of Mathematics Waterloo, Ontario N2L 3G Introduction Grade 7 & 8 Math Circles Circles, Circles, Circles March 9/20, 203 The circle is a very important shape. In fact of all shapes, the circle is

### Math 1B, lecture 5: area and volume

Math B, lecture 5: area and volume Nathan Pflueger 6 September 2 Introduction This lecture and the next will be concerned with the computation of areas of regions in the plane, and volumes of regions in

### Objective To introduce a formula to calculate the area. Family Letters. Assessment Management

Area of a Circle Objective To introduce a formula to calculate the area of a circle. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment

### Dŵr y Felin Comprehensive School. Perimeter, Area and Volume Methodology Booklet

Dŵr y Felin Comprehensive School Perimeter, Area and Volume Methodology Booklet Perimeter, Area & Volume Perimeters, Area & Volume are key concepts within the Shape & Space aspect of Mathematics. Pupils

### 10-3 Area of Parallelograms

0-3 Area of Parallelograms MAIN IDEA Find the areas of parallelograms. NYS Core Curriculum 6.A.6 Evaluate formulas for given input values (circumference, area, volume, distance, temperature, interest,

Think About This Situation A popular game held at fairs or parties is the jelly bean guessing contest. Someone fills a jar or other large transparent container with a known quantity of jelly beans and

### ACTIVITY: Finding a Formula Experimentally. Work with a partner. Use a paper cup that is shaped like a cone.

8. Volumes of Cones How can you find the volume of a cone? You already know how the volume of a pyramid relates to the volume of a prism. In this activity, you will discover how the volume of a cone relates

### MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

### Julie Rotthoff Brian Ruffles. No Pi Allowed

Julie Rotthoff Brian Ruffles rott9768@fredonia.edu ruff0498@fredonia.edu No Pi Allowed Introduction: Students often confuse or forget the area and circumference formulas for circles. In addition, students