Informal Argument for Volume of Sphere - Geometry

1. Identify the solid used with Cavalieri to derive $V=\frac{4}{3} \pi r^3$ for a sphere.

   A hemisphere compared to a cylinder minus a cone. Classic Cavalieri setup comparing hemisphere to cylinder minus cone.

2. Simplify the cross-section area $c^0r^2-c^0x^2$.

   $A=c^0(r^2-x^2)$. Factoring out common $\pi$ term from the difference.

3. What is the base radius of the cone in the classic hemisphere Cavalieri argument?

   $r$. Cone base matches hemisphere base for the Cavalieri comparison.

4. Identify the solid whose volume is $\frac{1}{3}$ of a cylinder with the same base and height.

   A cone. Cones have exactly one-third the volume of corresponding cylinders.

5. Compute $V_{\text{cone}}$ for base radius $r$ and height $r$.

   $V_{\text{cone}}=\frac{1}{3}c^0r^3$. Volume formula $V = \frac{1}{3}Bh = \frac{1}{3}\pi r^2 \cdot r$ for the cone.

6. Find $V_{\text{hemi}}$ from $V_{\text{cyl}}= \pi r^3$ and $V_{\text{cone}}= \frac{1}{3} \pi r^3$.

   $V_{\text{hemi}}= \frac{2}{3} \pi r^3$. Subtracting: $\pi r^3 - \frac{1}{3} \pi r^3 = \frac{2}{3} \pi r^3$.

7. What is the volume formula for a cone with base area $B$ and height $h$?

   $V=\frac{1}{3}Bh$. Standard formula where cones have $\frac{1}{3}$ the volume of corresponding prisms.

8. What is the volume of a sphere if its radius doubles from $r$ to $2r$?

   It becomes $8$ times as large: $V=\frac{4}{3}c^0(2r)^3$. Volume scales as the cube of radius: $(2r)^3 = 8r^3$.

9. Find $V_{\text{hemi}}$ from $V_{\text{cyl}}=c^0r^3$ and $V_{\text{cone}}=\frac{1}{3}c^0r^3$.

   $V_{\text{hemi}}=\frac{2}{3}c^0r^3$. Subtracting: $\pi r^3 - \frac{1}{3}\pi r^3 = \frac{2}{3}\pi r^3$.

10. Identify the solid used with Cavalieri to derive $V=\frac{4}{3}c^0r^3$ for a sphere.

    A hemisphere compared to a cylinder minus a cone. Classic Cavalieri setup comparing hemisphere to cylinder minus cone.

11. What condition on cross-sections is required to apply Cavalieri's principle?

    Equal cross-sectional areas at every corresponding height. Cross-sections must match exactly at every level for Cavalieri's principle to work.

12. What does it mean for two solids to have "congruent cross-sections" at height $h$?

    Their cross-sections have equal area at the same $h$. Cross-sections are congruent circles or shapes at each level.

13. What is the height of the cone in the classic hemisphere Cavalieri argument?

    $h=r$. Same as hemisphere radius to create matching cross-sections.

14. Which known volume formula is typically used as the starting point in Cavalieri arguments?

    $V=Bh$ for prisms and cylinders. This simple formula provides the foundation for more complex volume derivations.

15. What is the height of the cylinder in the classic hemisphere Cavalieri argument?

    $h=r$. Key dimension in the standard hemisphere Cavalieri argument.

16. What is the key idea in an informal Cavalieri argument for volume formulas?

    Match cross-sectional areas to a known solid at each height. Find a comparison solid with known volume and matching cross-sections.

17. What is the radius of the cylinder in the classic hemisphere Cavalieri argument?

    $r$. Matches the hemisphere radius for the comparison to work.

18. What must be true about the heights of two solids to use Cavalieri's principle?

    They must have the same height (same range of $h$ values). Both solids must span identical vertical ranges for valid comparison.

19. Which solid has constant cross-sectional area $A(h)=B$ for all heights $h$?

    A prism (including a cylinder). Uniform cross-sectional area characterizes prisms and cylinders.

20. State the relationship between volumes if $A_1(h)=kA_2(h)$ for all heights $h$.

    $V_1=kV_2$. If cross-section areas have constant ratio $k$, volumes have same ratio.

21. What is the cross-section area of (cylinder minus cone) at height $x$ from the base, with $r$ and $h=r$?

    $A=c^0r^2-c^0x^2$. Cylinder area minus cone area at the same height.

22. What is the cross-section area of a cylinder of radius $r$ at any height?

    $A = \pi r^2$. Uniform circular cross-section for any cylinder.

23. Which equality of cross-sectional areas supports the hemisphere Cavalieri argument at height $x$?

    $c^0(r^2-x^2)=c^0r^2-c^0x^2$. This equality proves hemisphere matches cylinder minus cone cross-sections.

24. What is the volume formula for a pyramid with base area $B$ and height $h$?

    $V=\frac{1}{3}Bh$. Standard formula where pyramids have $\frac{1}{3}$ the volume of corresponding prisms.

25. What volume does Cavalieri give for a hemisphere if it matches cylinder minus cone?

    $V_{\text{hemi}}=V_{\text{cyl}}-V_{\text{cone}}$. Cavalieri's principle applied - equal cross-sections give equal volumes.

26. What is Cavalieri's principle for comparing volumes of two solids?

    If all cross-sections have equal area at every height, volumes are equal. This is the fundamental statement of Cavalieri's principle for volume comparison.

27. What is the cross-section area of a hemisphere of radius $r$ at height $x$ above the base?

    $A = \pi (r^2 - x^2)$. Circular cross-section area using Pythagorean theorem for radius.

28. Find $V_{\text{sphere}}$ if $V_{\text{hemi}}=\frac{2}{3}\pi r^3$.

    $V_{\text{sphere}}=\frac{4}{3}\pi r^3$. Double the hemisphere volume: $2 \times \frac{2}{3} \pi r^3$.

29. What is the cross-section area of that cone at height $x$ from the apex?

    $A=c^0x^2$. Cross-sectional area grows quadratically with distance from apex.

30. Identify the solid whose volume is $\frac{1}{3}$ of a prism with the same base and height.

    A pyramid. Pyramids have exactly $\frac{1}{3}$ the volume of corresponding prisms.

Study these flashcards