3D figures (Cavalieri's principle) Knowledge-Based Study

Study notes

1. What is Cavalieri's Principle?

   Cavalieri's Principle is a foundational geometric concept formulated by the Italian mathematician Bonaventura Cavalieri in the 17th century. It allows us to compare and determine the volumes of solids without relying on complex integration.

   Intuitive Foundation: A Stack of Coins

   Consider a neat, vertical stack of identical circular coins. This stack forms a right circular cylinder. If you gently push the stack to one side so that it leans, it forms an oblique cylinder. Although the shape has shifted, the total amount of metal in the stack has not changed. Each horizontal slice (a single coin) maintains the exact same area, and the total vertical height from the bottom coin to the top coin remains identical.

   Formal Statement for Three-Dimensional Solids

   If two solids have the same height $h$ and if every plane parallel to the bases cuts both solids in cross-sections of equal area, then the two solids have equal volumes.

   • Base planes: The solids must lie between two parallel planes.
   • Cross-sectional equality: At every intermediate height $z$, the cross-sectional area $A_1(z)$ of the first solid must equal the cross-sectional area $A_2(z)$ of the second solid: $A_1(z) = A_2(z)$.
   • Conclusion: $\text{Volume}_1 = \text{Volume}_2$.

   Volume of Oblique Prisms and Cylinders

   Cavalieri's principle provides immediate proof that the standard volume formulas for prisms and cylinders apply whether the figure is right or oblique.

   Comparing Right and Oblique Cylinders

   Let a right circular cylinder and an oblique circular cylinder both have base radius $r$ and vertical height $h$.

   • Any plane parallel to the base cuts the right cylinder in a circle of area $A = \pi r^2$.
   • Because the lateral boundary shifts linearly without compressing or stretching the horizontal cross-section, the same parallel plane cuts the oblique cylinder in a circle of identical area $A = \pi r^2$.

   Since the cross-sectional areas match at every vertical level across the entire height $h$, Cavalieri's principle guarantees:

   $$V = \text{Base Area} \times \text{Height} = B \cdot h = \pi r^2 h$$

   Key Takeaway: When calculating the volume of an oblique prism or cylinder, always use the perpendicular (vertical) height, not the slant length of the lateral edges.

   Pyramids and Cones with Equal Cross-Sections

   Cavalieri's principle extends naturally to tapering solids, such as cones and pyramids.

   Cross-Sections at Height $y$

   Consider two pyramids of the same height $h$, whose bases have the same area $B$, regardless of whether their bases are congruent polygons or have different shapes (for example, a triangle and a pentagon).

   By similar figures, a cross-section cut at a distance $y$ below the apex has a linear scale factor of $\frac{y}{h}$ relative to the base. Therefore, its area is scaled by the square of this ratio:

   $$A(y) = B \left(\frac{y}{h}\right)^2$$

   Because this cross-sectional formula depends only on the base area $B$, the total height $h$, and the distance $y$ from the apex:

   • Two pyramids with equal base areas and equal heights have identical cross-sectional areas at every horizontal plane.
   • A cone and a pyramid with equal base areas ($B = \pi r^2$) and equal heights have identical cross-sectional areas at every level.

   Therefore, by Cavalieri's principle, every such pyramid or cone has the exact same volume:

   $$V = \frac{1}{3} B h$$

   Cavalieri's Classic Proof: The Volume of a Sphere

   One of the most famous historical applications of Cavalieri's principle is the derivation of the volume of a sphere of radius $R$.

   Constructing the Comparison Solids

   Consider two shapes positioned between the horizontal planes $z = 0$ and $z = R$:

   1. Solid 1: A hemisphere of radius $R$.
   2. Solid 2: A right circular cylinder of radius $R$ and height $R$, from which an inverted right circular cone of base radius $R$ and height $R$ has been hollowed out.

   Evaluating Cross-Sections at Height $z$

   Let a plane slice through both solids at a height $z$ above the base ($0 \le z \le R$):

   • In the hemisphere, the cross-section is a circle with radius $r(z) = \sqrt{R^2 - z^2}$ by the Pythagorean theorem. Its area is:$$A_{\text{hemisphere}}(z) = \pi \left(\sqrt{R^2 - z^2}\right)^2 = \pi (R^2 - z^2)$$
   • In the hollowed cylinder, the cross-section is an annulus (a ring). The outer circle has radius $R$, and the inner circular hole cut by the cone has radius $z$ (since the cone has a 1-to-1 slope ratio). Its area is:$$A_{\text{annulus}}(z) = \pi R^2 - \pi z^2 = \pi (R^2 - z^2)$$

   Since $A_{\text{hemisphere}}(z) = A_{\text{annulus}}(z)$ at every height $z$, Cavalieri's principle confirms that their volumes are equal:

   $$V_{\text{hemisphere}} = V_{\text{cylinder}} - V_{\text{cone}} = \pi R^2(R) - \frac{1}{3}\pi R^2(R) = \frac{2}{3}\pi R^3$$

   Multiplying by 2 yields the volume of the entire sphere: $V_{\text{sphere}} = \frac{4}{3}\pi R^3$.

   Summary of Key Conditions

   To successfully apply Cavalieri's principle, ensure the following conditions are met:

   • Both solids must have the same total vertical height.
   • The cross-sections must be taken parallel to the base at equal vertical levels.
   • The cross-sections must have equal areas at all levels, although their geometric shapes need not be identical.

   Worked Example

   Problem: An oblique triangular prism has a vertical height of 12 cm and a base triangle with a base of 6 cm and an altitude of 5 cm. A right rectangular prism has a base measuring 3 cm by 5 cm and a vertical height of 12 cm. Do the two prisms have equal volumes?

   Solution:

   1. Calculate the base area of the triangular prism:$$B_1 = \frac{1}{2} \cdot 6 \cdot 5 = 15\text{ cm}^2$$
   2. Calculate the base area of the rectangular prism:$$B_2 = 3 \cdot 5 = 15\text{ cm}^2$$
   3. Check cross-sections: Because every horizontal slice of a prism is congruent to its base, every slice of the triangular prism has an area of $15\text{ cm}^2$, and every slice of the rectangular prism has an area of $15\text{ cm}^2$.
   4. Check heights: Both solids have an identical vertical height of 12 cm.

   By Cavalieri's principle, both prisms have equal volumes: $V = 15 \cdot 12 = 180\text{ cm}^3$.