# Unit 1: Circles & volume

WARM UP! TODAYS AGENDA WARM UP SYLLA BUS FORMULA SH EET CA LCULATOR FORM LEA RN IN G!! !

H OMEWORK UNIT 1: CIRCLES & VOLUME TEST 1/24 CONCEPTS/SKILLS TO MAINTAIN

The introduction to all of the parts of a circle and the relationships of all of those parts to each other will be new to you this semester. The concepts of Area, Surface Area, and Volume of triangles, special quadrilaterals, and right rectangular prisms were introduced in the 6th Grade Unit 5. This knowledge was built on in the 7th Grade Unit 5 and expanded to include the

slicing of right rectangular pyramids. The Volumes of Cones, Cylinders, and Spheres were previously covered in the 8 th Grade Unit 3. The purpose of re-visiting these formulas here in Analytic Geometry is to formalize your understanding of the Unit 1: Circles & volume L G 1 - 1 V O LU M E

L G 1 - 2 T H E RAD I A N L G 1 - 3 C I RC L E THEOREMS LEARNING GOAL 1-1 QUESTION: How does Cavalieris Principle apply to finding the volume of a cylinder, even if it

is oblique or not standing straight up? ESSENTIAL SKILLS: Explain Volume Formulas and Use them to solve problems Extend the study of identifying cross sections of three dimensional shapes to SELECTED TERMS AND

SYMBOLS Cavalieris Principle: A method, with formula given below, of finding the volume of any solid for which cross-sections by parallel planes have equal areas. This includes, but is not limited to, cylinders and prisms. Composite Figures: If a figure is made from two or more geometric figures, then it is called a Composite Figure.

Lateral Area: The sum of the areas of the lateral ( vertical) faces of a cylinder, cone, frustum or the like. Slant Height: The diagonal distance from the apex of 2 2hwh 2lh of l24Area

rlw r2w r2rh23h 24base r triangles sr h H Area of base 3

3 VOLUME FORMULAS MORE VOLUME FORMULAS MIXED VOLUME Complete the Mixed Volume worksheet. Whatever you dont complete in class is your

homework WARM UP! COMPOSITE SOLID VOLUME Composite solids are made up of more than one standard 3D shape.

VOLUME APPLICATIONS Complete the worksheet on Volume Applications. Finish this for homework WARM UP! THIS IS SPARTA! POWE RED BY P RIDE!

1)Commitment 2)Cooperation 3)Integrity 4)Respect 5)Responsibility CLASSROOM BEHAVIOR EXPECTATIONS Students should:

Be on time Actively participate in class Follow adult direction Work collaboratively with peers Use appropriate language; respond appropriately Take ownership of actions HALLWAY BEHAVIOR

EXPECTATIONS Students should: Walk with purpose Keep right One ear bud rule Be respectful and responsible IMPORTANT TO REMEMBER:

Many teachers have their own set of Classroom Behavior Rules or Expectations. These expectations should be followed in addition to any other rules or expectations in your teachers class. From this point on you will be held accountable for following all of these expectations. Consider this your warning

CAVALIERIS PRINCIPLE Volumes of Cylinders, Cones, Pyramids, and Spheres Task VOLUME OF CYLINDERS AND CONES What is the difference in volume between cylinders and cones with the same area bases and same height? How do you find the formula for the volume of a

cylinder? How do you find the formula for the volume of a cone? Volume Videos: Volume of a Cone Volume of a Pyramid Volume of a Sphere WARM UP!

RESTROOM EXPECTATIONS Have a pass Be quick and quiet Clean up Be respectful and responsible LUNCH ROOM EXPECTATIONS Report and remain

Be quick and quiet. Clean area Be respectful and responsible LEARNING COMMONS EXPECTATIONS Have a pass Sign in Work quietly

Be respectful and responsible VOLUME COMPARISON The radius of the earth is approximately 3950 miles. The radius of Jupiter is approximately 44,400 miles. Since the radius of Jupiter is more than 11 times that of the earth, which comes closest to the volume of Jupiter ? a) About 11 times that of the earth

b) About 121 times that of the earth c) About 600 times that of the earth d. About 1300 times that of the earth TRUE OR FALSE? 1) If you double just the width of a rectangular prism then you double its volume. 2) If you double just the width and height of a rectangular prism then you double its volume.

3) If you double the width, height, and length of a rectangular prism then you double its volume. 4) If you double the radius of a circle then you double its circumference. 5) If you double the radius of a circle then you double its area. 6) If you double the radius of a sphere then you double CAVALIERIS PRINCIPLE

The volumes of two objects of the same height are equal if the areas of their corresponding cross sections are equal. CAVALIERIS PRINCIPLE EXAMPLES These pieces maintain their

SAME volume regardless of how they are moved. CAVALIERIS PRINCIPLE EXAMPLES All heights are the same. Since each cross sections area are congruent, all the volumes are equal.

CAVALIERIS PRINCIPLE The same volume formula applies EXAMPLES whether its a right prism or an oblique prism. V = Bh

EXAMPLE 1: WHICH SOLID HAS MORE VOLUME? USE CAVALIERIS PRINCIPLE. 20 in. 20 in. 4 in. 6 in.

8 in. 6 in. The area of the cross sections are both 24in2 and have the same height of 20, so the solids have the SAME volume.

EXAMPLE 2: WHICH SOLID HAS MORE VOLUME? USE CAVALIERIS PRINCIPLE. 6 cm 10 cm 6 cm

The area of the cross sections are both 36cm2 and have the same height of 10, so the solids have the SAME volume. QUEEEEEEZ TIME! You have the remainder of class to finish the quiz. When you have selected your answers, raise your hand and I will

check your answers. If I fold your paper in half, you have earned 100% on the quiz. Congrats! Youre done tap a nap! If you have missed any questions, I will tell you the number of questions that you missed. The number I give you does not correspond to the question number. Example: I do not fold your paper and instead say One to you. This means you have missed one question and it is YOUR job to find the incorrect answer and fix it.

Once you feel you have found and corrected your answer, raise your hand again and I will check again. At this point, it is STILL POSSIBLE to make 100% on the quiz! If I now

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