Cavalieri's Principle and Cross Sections

MAFS.912.G-GMD.1.1 Practice Lesson

MAFS.912.G-GMD.1.1

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments

Finding side length of a parallelogram when given a sectioned circle

•  $Circumference=2\pi r$  First you need to find the circumference

•  $Area\ of\ a\ circle=2\pi r^2$  You will need this formula later

Finding side length cont...

• Next you need to divide the circumference by 2

• Why?

• We divide the circumference by 2 because half of the circle makes up each base of the parallelogram.

Cavalieri's Principle

If two solids of the same height are such that their corresponding sections at any level have the same areas, then they have the same volume.

Multiple Choice

Cavalieri's Principle is most closely hinged on which measurements?

• Height and area

• Mass and force

• Pi and diameter

• Radius and pi

Cavalieri's Principle

These solids are clearly different, but they have the same height, and the area of the cross sections are equal when the solids are cut, so they have equal volume.

Multiple Choice

Would these two solids satisfy Cavalieri's principle?

• Yes!

• No!

Multiple Choice

What are the volumes of cylinder A and B?

• 110π in³

• 220π in³

• 440π in³

• 550π in³

• They do not have the same volume.