Slide 1

DRILL(A) Name the cross-sections you would find in a cone, cylinder, cube, rectangular prism.(B) What solids would you use to model these farm structures? Why would we want to know?

1What would you get if you turned these shapes about their axes?Axis bisects triangleRotating Triangle in 3D

Rotation creates a cone

Core Lesson3Edge along axis forms center axis of solidTriangle: Axis Along EdgeOther edges create curved surfaces

Core Lesson4

Edges perpendicular to axis draw flat facesRectangle: Axis BisectingEdges parallel to axis draw curved surfacesRotation creates: cylinder

Core Lesson5

Edges perpendicular to axis draw flat facesRectangle: Axis Along EdgeEdges parallel to axis draw curved surfacesRotation creates: cylinder

Core Lesson6

Curved edges draw curved surfacesCircle: Axis BisectingRotation creates: sphere

Core Lesson7

Circle: Axis Along EdgeCurved edges draw curved surfacesRotation creates: torus

Core Lesson8

What is VOLUME?

Core LessonWhy does V = B x h calculate the volume of prisms & cylinders?How do you know you can trust the formulas?V = B x hBh

Why does A = r2 calculate the area of a circle? After all, this seems like a magic formula. But how do you know you can trust it?

10Cavalieri PrincipleBonaventuraCavalieri

If cross-sectional area of two prisms is the same for every height above the base, then the volumes will be the same.

Core LessonLets start with our circle. Slice it up into equal sections, like really large slices of pizza. Now rearrange those slices. We get this bumpy shape that very vaguely resembles a parallelogram. Lets look at what our measurements are.11Cavalieris Principle

Core LessonB = 2.86in2 h = .7in

Cylinder: US QuarterCore LessonLets start with our circle. Slice it up into equal sections, like really large slices of pizza. Now rearrange those slices. We get this bumpy shape that very vaguely resembles a parallelogram. Lets look at what our measurements are.13B = 2.86in2 h = 11.2 inStack of 16 quarters

V = 2.86 x 11.2 = 32 in3 Core LessonLets start with our circle. Slice it up into equal sections, like really large slices of pizza. Now rearrange those slices. We get this bumpy shape that very vaguely resembles a parallelogram. Lets look at what our measurements are.14Works for unusual shapes

If base area is congruent, multiply B x h to easily calculate volume.heightCore LessonLets start with our circle. Slice it up into equal sections, like really large slices of pizza. Now rearrange those slices. We get this bumpy shape that very vaguely resembles a parallelogram. Lets look at what our measurements are.15Right & Oblique Prisms & Cylinders

Core Lesson