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?

What would you get if you turned these shapes about their axes?

Core Lesson Axis bisects triangle Rotating Triangle in 3D Rotation creates a cone

Core Lesson Edge along axis forms center axis of solid Triangle: Axis Along Edge Other edges create curved surfaces

Core Lesson Edges perpendicular to axis draw flat faces Rectangle: Axis Bisecting Edges parallel to axis draw curved surfaces Rotation creates: cylinder

Core Lesson Edges perpendicular to axis draw flat faces Rectangle: Axis Along Edge Edges parallel to axis draw curved surfaces Rotation creates: cylinder

Core Lesson Curved edges draw curved surfaces Circle: Axis Bisecting Rotation creates: sphere

Core Lesson Circle: Axis Along Edge Curved edges draw curved surfaces Rotation creates: torus

Core Lesson What is VOLUME?

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

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

Core Lesson Cavalieri’s Principle

Core Lesson B = 2.86in 2 h =.7in Cylinder: US Quarter

Core Lesson B = 2.86in 2 h = 11.2 in Stack of 16 quarters V = 2.86 x 11.2 = 32 in 3

Core Lesson Works for unusual shapes If base area is congruent, multiply B x h to easily calculate volume. heigh t

Core Lesson Right & Oblique Prisms & Cylinders