Cavalieri's Principle

Cavalieri's Principle and Cross Sections

Thinks about it.....​

Do the 2 stacks of coins have the same volume?

What is a cross section?

A cross-section is the shape we get when cutting through an object

When cutting a cylinder parallel to the base, the cross-section is a circle.

What is the cross section of a cylinder when cutting it perpendicular to the base?

The perpendicular cross section of a cylinder is a rectangle.

What happens to the rectangle cross-sections as you move closer to the edge of the cylinder?

The cross-section of a rectangular pyramid parallel to the base is a rectangle.

What relationship does a cross-section parallel to the base have with the base?

Multiple Choice

Describe the cross-section.

• square

• rectangle

• triangle

• circle

Multiple Choice

What shape will the vertical cross -section of a cylinder be?

• Circle

• Triangle

• Rectangle

• Ellipse

Multiple Choice

Describe the cross section.

• triangle
• rectangle
• trapezoid
• square

Multiple Choice

What shape will the horizontal cross-section of a sphere be?

• Dome

• Circle

• Ellipse

• Rectangle

Do the two stacks of coins have the same volume? How do you know?

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Since the 2 stacks have the same kind and amount of coins, it makes sense to say that the two stacks have the same volume. The arrangement of the coins does not effect the volume. This is Cavelieri's Principle. The coins have the same cross-sectional area (same coin) at every plane parallel to the base.

Compare the volume of the stacks of coins.

• Some of the coins are larger than the others. There is no easy way to tell whether the stacks have the same volume.

Compare the volume of the pyramids.

• The volumes of the pyramids do not change.

• Cavalieri's Principal can help explain why.

Cavalieri's Principle

https://schoolyourself.org/learn/geometry/cavalieri-3d

Multiple Choice

An oblique shape will always have the same volume as a right shape if they have the same base.

• True

• False - an oblique shape and a right shape can never have the same volume

• False - an oblique shape and a right shape can't have the same base

• False - an oblique shape will always have the same volume as a right shape if they have the same base AND height

Multiple Choice

The shapes below have the Base Areas and Heights. Which will have the same volume. Which ones will have the same volumes based on Cavalieri's Principle?

• Rectangular Pyramid and Triangular Prism

• Rectangular Pyramid and Cone

• Triangular Prism and Cone

• All three.

Multiple Choice

The shapes below have the Base Areas and Heights. Which will have the same volume. Which ones will have the same volumes based on Cavalieri's Principle?

• Rectangular Pyramid and Triangular Prism

• Rectangular Pyramid and Cylinder

• Triangular Prism and Cylinder

• All three.

Multiple Choice

Based on Cavalieri's Principle, will the two prisms have the same volume?

• No, they will not be same. Although the heights are the same, the cross-sections are different shapes. 

• Yes, the heights of both prisms are the same and they have the same cross-sectional area. Therefore, they will have the same volume.

Multiple Choice

Jenny says that the two prisms DO NOT have the same volume because the cross sections are not the same. Renee disagrees; she says that it isn't the shape that has to be the same but the area. Renee thinks they have the same volume.

Who is correct?

• Jenny

• Renee

• Neither Jenny or Renee is correct.

Multiple Choice

Determine whether the 3-D figures are congruent to each other. (Hint: Find the area of the cross-sectional shapes).

• False. Although the cross-sectionals' area and heights of the 3-D figures are the same,the shapes are not congruent

• True. The cross-sectionals' area and the heights of the 3-D figures are the same. They don't have to be the same shape.

• False. The cross-sections are not the same shape.

• True. All the shapes are 3-D figures and the same height.

Multiple Choice

a) Cavalieri’s Principle states that any two objects with the same cross-sectional areas and heights must have the same volume.

• True

• False - the cross sectional areas are not relevant

• False - only the slant height is relevant

• False - even if they have the same cross sectional areas and heights, they cannot have the same volume.