Informal Argument for Volume of Sphere - Geometry
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What is Cavalieri's principle for comparing volumes of two solids?
What is Cavalieri's principle for comparing volumes of two solids?
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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.
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.
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What condition on cross-sections is required to apply Cavalieri's principle?
What condition on cross-sections is required to apply Cavalieri's principle?
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Equal cross-sectional areas at every corresponding height. Cross-sections must match exactly at every level for Cavalieri's principle to work.
Equal cross-sectional areas at every corresponding height. Cross-sections must match exactly at every level for Cavalieri's principle to work.
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What must be true about the heights of two solids to use Cavalieri's principle?
What must be true about the heights of two solids to use Cavalieri's principle?
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They must have the same height (same range of $h$ values). Both solids must span identical vertical ranges for valid comparison.
They must have the same height (same range of $h$ values). Both solids must span identical vertical ranges for valid comparison.
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What is the volume formula for a cone with base area $B$ and height $h$?
What is the volume formula for a cone with base area $B$ and height $h$?
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$V= rac{1}{3}Bh$. Standard formula where cones have $\frac{1}{3}$ the volume of corresponding prisms.
$V= rac{1}{3}Bh$. Standard formula where cones have $\frac{1}{3}$ the volume of corresponding prisms.
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What is the volume formula for a pyramid with base area $B$ and height $h$?
What is the volume formula for a pyramid with base area $B$ and height $h$?
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$V= rac{1}{3}Bh$. Standard formula where pyramids have $\frac{1}{3}$ the volume of corresponding prisms.
$V= rac{1}{3}Bh$. Standard formula where pyramids have $\frac{1}{3}$ the volume of corresponding prisms.
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What is the height of the cylinder in the classic hemisphere Cavalieri argument?
What is the height of the cylinder in the classic hemisphere Cavalieri argument?
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$h=r$. Key dimension in the standard hemisphere Cavalieri argument.
$h=r$. Key dimension in the standard hemisphere Cavalieri argument.
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What is the radius of the cylinder in the classic hemisphere Cavalieri argument?
What is the radius of the cylinder in the classic hemisphere Cavalieri argument?
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$r$. Matches the hemisphere radius for the comparison to work.
$r$. Matches the hemisphere radius for the comparison to work.
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What is the height of the cone in the classic hemisphere Cavalieri argument?
What is the height of the cone in the classic hemisphere Cavalieri argument?
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$h=r$. Same as hemisphere radius to create matching cross-sections.
$h=r$. Same as hemisphere radius to create matching cross-sections.
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What is the base radius of the cone in the classic hemisphere Cavalieri argument?
What is the base radius of the cone in the classic hemisphere Cavalieri argument?
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$r$. Cone base matches hemisphere base for the Cavalieri comparison.
$r$. Cone base matches hemisphere base for the Cavalieri comparison.
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What is the cross-section area of that cone at height $x$ from the apex?
What is the cross-section area of that cone at height $x$ from the apex?
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$A=c^0x^2$. Cross-sectional area grows quadratically with distance from apex.
$A=c^0x^2$. Cross-sectional area grows quadratically with distance from apex.
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What is the cross-section area of a cylinder of radius $r$ at any height?
What is the cross-section area of a cylinder of radius $r$ at any height?
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$A = \pi r^2$. Uniform circular cross-section for any cylinder.
$A = \pi r^2$. Uniform circular cross-section for any cylinder.
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What is the cross-section area of (cylinder minus cone) at height $x$ from the base, with $r$ and $h=r$?
What is the cross-section area of (cylinder minus cone) at height $x$ from the base, with $r$ and $h=r$?
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$A=c^0r^2-c^0x^2$. Cylinder area minus cone area at the same height.
$A=c^0r^2-c^0x^2$. Cylinder area minus cone area at the same height.
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Simplify the cross-section area $c^0r^2-c^0x^2$.
Simplify the cross-section area $c^0r^2-c^0x^2$.
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$A=c^0(r^2-x^2)$. Factoring out common $\pi$ term from the difference.
$A=c^0(r^2-x^2)$. Factoring out common $\pi$ term from the difference.
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Which equality of cross-sectional areas supports the hemisphere Cavalieri argument at height $x$?
Which equality of cross-sectional areas supports the hemisphere Cavalieri argument at height $x$?
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$c^0(r^2-x^2)=c^0r^2-c^0x^2$. This equality proves hemisphere matches cylinder minus cone cross-sections.
$c^0(r^2-x^2)=c^0r^2-c^0x^2$. This equality proves hemisphere matches cylinder minus cone cross-sections.
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What volume does Cavalieri give for a hemisphere if it matches cylinder minus cone?
What volume does Cavalieri give for a hemisphere if it matches cylinder minus cone?
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$V_{ ext{hemi}}=V_{ ext{cyl}}-V_{ ext{cone}}$. Cavalieri's principle applied - equal cross-sections give equal volumes.
$V_{ ext{hemi}}=V_{ ext{cyl}}-V_{ ext{cone}}$. Cavalieri's principle applied - equal cross-sections give equal volumes.
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Compute $V_{ ext{cone}}$ for base radius $r$ and height $r$.
Compute $V_{ ext{cone}}$ for base radius $r$ and height $r$.
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$V_{ ext{cone}}= rac{1}{3}c^0r^3$. Volume formula $V = \frac{1}{3}Bh = \frac{1}{3}\pi r^2 \cdot r$ for the cone.
$V_{ ext{cone}}= rac{1}{3}c^0r^3$. Volume formula $V = \frac{1}{3}Bh = \frac{1}{3}\pi r^2 \cdot r$ for the cone.
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What is the key idea in an informal Cavalieri argument for volume formulas?
What is the key idea in an informal Cavalieri argument for volume formulas?
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Match cross-sectional areas to a known solid at each height. Find a comparison solid with known volume and matching cross-sections.
Match cross-sectional areas to a known solid at each height. Find a comparison solid with known volume and matching cross-sections.
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Which known volume formula is typically used as the starting point in Cavalieri arguments?
Which known volume formula is typically used as the starting point in Cavalieri arguments?
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$V=Bh$ for prisms and cylinders. This simple formula provides the foundation for more complex volume derivations.
$V=Bh$ for prisms and cylinders. This simple formula provides the foundation for more complex volume derivations.
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What does it mean for two solids to have “congruent cross-sections” at height $h$?
What does it mean for two solids to have “congruent cross-sections” at height $h$?
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Their cross-sections have equal area at the same $h$. Cross-sections are congruent circles or shapes at each level.
Their cross-sections have equal area at the same $h$. Cross-sections are congruent circles or shapes at each level.
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Identify the solid whose volume is $ rac{1}{3}$ of a cylinder with the same base and height.
Identify the solid whose volume is $ rac{1}{3}$ of a cylinder with the same base and height.
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A cone. Cones have exactly one-third the volume of corresponding cylinders.
A cone. Cones have exactly one-third the volume of corresponding cylinders.
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State the relationship between volumes if $A_1(h)=kA_2(h)$ for all heights $h$.
State the relationship between volumes if $A_1(h)=kA_2(h)$ for all heights $h$.
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$V_1=kV_2$. If cross-section areas have constant ratio $k$, volumes have same ratio.
$V_1=kV_2$. If cross-section areas have constant ratio $k$, volumes have same ratio.
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Which solid has constant cross-sectional area $A(h)=B$ for all heights $h$?
Which solid has constant cross-sectional area $A(h)=B$ for all heights $h$?
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A prism (including a cylinder). Uniform cross-sectional area characterizes prisms and cylinders.
A prism (including a cylinder). Uniform cross-sectional area characterizes prisms and cylinders.
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What is the volume of a sphere if its radius doubles from $r$ to $2r$?
What is the volume of a sphere if its radius doubles from $r$ to $2r$?
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It becomes $8$ times as large: $V= rac{4}{3}c^0(2r)^3$. Volume scales as the cube of radius: $(2r)^3 = 8r^3$.
It becomes $8$ times as large: $V= rac{4}{3}c^0(2r)^3$. Volume scales as the cube of radius: $(2r)^3 = 8r^3$.
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Find $V_{ ext{hemi}}$ from $V_{ ext{cyl}}= \pi r^3$ and $V_{ ext{cone}}= \frac{1}{3} \pi r^3$.
Find $V_{ ext{hemi}}$ from $V_{ ext{cyl}}= \pi r^3$ and $V_{ ext{cone}}= \frac{1}{3} \pi r^3$.
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$V_{ ext{hemi}}= \frac{2}{3} \pi r^3$. Subtracting: $\pi r^3 - \frac{1}{3} \pi r^3 = \frac{2}{3} \pi r^3$.
$V_{ ext{hemi}}= \frac{2}{3} \pi r^3$. Subtracting: $\pi r^3 - \frac{1}{3} \pi r^3 = \frac{2}{3} \pi r^3$.
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Identify the solid whose volume is $ rac{1}{3}$ of a prism with the same base and height.
Identify the solid whose volume is $ rac{1}{3}$ of a prism with the same base and height.
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A pyramid. Pyramids have exactly $ rac{1}{3}$ the volume of corresponding prisms.
A pyramid. Pyramids have exactly $ rac{1}{3}$ the volume of corresponding prisms.
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Find $V_{ ext{sphere}}$ if $V_{ ext{hemi}}=\frac{2}{3}\pi r^3$.
Find $V_{ ext{sphere}}$ if $V_{ ext{hemi}}=\frac{2}{3}\pi r^3$.
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$V_{ ext{sphere}}=\frac{4}{3}\pi r^3$. Double the hemisphere volume: $2 \times \frac{2}{3} \pi r^3$.
$V_{ ext{sphere}}=\frac{4}{3}\pi r^3$. Double the hemisphere volume: $2 \times \frac{2}{3} \pi r^3$.
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What is the cross-section area of a hemisphere of radius $r$ at height $x$ above the base?
What is the cross-section area of a hemisphere of radius $r$ at height $x$ above the base?
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$A = \pi (r^2 - x^2)$. Circular cross-section area using Pythagorean theorem for radius.
$A = \pi (r^2 - x^2)$. Circular cross-section area using Pythagorean theorem for radius.
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Identify the solid used with Cavalieri to derive $V=\frac{4}{3} \pi r^3$ for a sphere.
Identify the solid used with Cavalieri to derive $V=\frac{4}{3} \pi r^3$ for a sphere.
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A hemisphere compared to a cylinder minus a cone. Classic Cavalieri setup comparing hemisphere to cylinder minus cone.
A hemisphere compared to a cylinder minus a cone. Classic Cavalieri setup comparing hemisphere to cylinder minus cone.
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What is the volume formula for a cone with base area $B$ and height $h$?
What is the volume formula for a cone with base area $B$ and height $h$?
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$V=\frac{1}{3}Bh$. Standard formula where cones have $\frac{1}{3}$ the volume of corresponding prisms.
$V=\frac{1}{3}Bh$. Standard formula where cones have $\frac{1}{3}$ the volume of corresponding prisms.
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Find $V_{ ext{hemi}}$ from $V_{ ext{cyl}}=c^0r^3$ and $V_{ ext{cone}}= rac{1}{3}c^0r^3$.
Find $V_{ ext{hemi}}$ from $V_{ ext{cyl}}=c^0r^3$ and $V_{ ext{cone}}= rac{1}{3}c^0r^3$.
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$V_{ ext{hemi}}= rac{2}{3}c^0r^3$. Subtracting: $\pi r^3 - \frac{1}{3}\pi r^3 = \frac{2}{3}\pi r^3$.
$V_{ ext{hemi}}= rac{2}{3}c^0r^3$. Subtracting: $\pi r^3 - \frac{1}{3}\pi r^3 = \frac{2}{3}\pi r^3$.
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