Informal Arguments for Circle/Solid Formulas - Geometry
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Find the area of a circle with diameter $d = 10$.
Find the area of a circle with diameter $d = 10$.
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$25\pi$. Use $r = 5$ from $d = 10$, then $A = \pi r^2$.
$25\pi$. Use $r = 5$ from $d = 10$, then $A = \pi r^2$.
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What is Cavalieri's principle stated informally for two solids with equal heights?
What is Cavalieri's principle stated informally for two solids with equal heights?
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Equal cross-sectional areas at every height imply equal volumes. Fundamental principle for comparing volumes using cross-sections.
Equal cross-sectional areas at every height imply equal volumes. Fundamental principle for comparing volumes using cross-sections.
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State the formula for the volume of a cone with base radius $r$ and height $h$.
State the formula for the volume of a cone with base radius $r$ and height $h$.
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$V = \frac{1}{3}\pi r^2 h$. Cones have one-third the volume of matching cylinders.
$V = \frac{1}{3}\pi r^2 h$. Cones have one-third the volume of matching cylinders.
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State the formula for the volume of a pyramid with base area $B$ and height $h$.
State the formula for the volume of a pyramid with base area $B$ and height $h$.
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$V = \frac{1}{3}Bh$. Pyramids have one-third the volume of matching prisms.
$V = \frac{1}{3}Bh$. Pyramids have one-third the volume of matching prisms.
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State the general volume relationship for a prism-like solid using base area $B$ and height $h$.
State the general volume relationship for a prism-like solid using base area $B$ and height $h$.
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$V = Bh$. General formula for prisms and cylinders.
$V = Bh$. General formula for prisms and cylinders.
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Identify the cross-sectional area used to justify cylinder volume by stacking identical slices.
Identify the cross-sectional area used to justify cylinder volume by stacking identical slices.
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Each cross-section is a circle of area $\pi r^2$. Stack circular disks of constant area $\pi r^2$.
Each cross-section is a circle of area $\pi r^2$. Stack circular disks of constant area $\pi r^2$.
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State the formula for the volume of a right circular cylinder with radius $r$ and height $h$.
State the formula for the volume of a right circular cylinder with radius $r$ and height $h$.
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$V = \pi r^2 h$. Volume equals base area times height for a cylinder.
$V = \pi r^2 h$. Volume equals base area times height for a cylinder.
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What informal limit statement explains why the sector-dissection shape approaches a true rectangle?
What informal limit statement explains why the sector-dissection shape approaches a true rectangle?
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As sector count increases, curved edges become nearly straight. More sectors create smoother, straighter rectangle edges.
As sector count increases, curved edges become nearly straight. More sectors create smoother, straighter rectangle edges.
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Using the dissection rectangle idea, what product gives the circle area?
Using the dissection rectangle idea, what product gives the circle area?
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$A = (\pi r)(r)$. Rectangle area formula: base times height gives $\pi r^2$.
$A = (\pi r)(r)$. Rectangle area formula: base times height gives $\pi r^2$.
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What is the area of a quarter circle with radius $r$ in terms of $\pi$?
What is the area of a quarter circle with radius $r$ in terms of $\pi$?
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$\frac{1}{4}\pi r^2$. One-fourth the area of a full circle with radius $r$.
$\frac{1}{4}\pi r^2$. One-fourth the area of a full circle with radius $r$.
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Find the circumference of a circle with radius $r = 5$.
Find the circumference of a circle with radius $r = 5$.
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$10\pi$. Apply $C = 2\pi r$ with $r = 5$.
$10\pi$. Apply $C = 2\pi r$ with $r = 5$.
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Find the circumference of a circle with diameter $d = 12$.
Find the circumference of a circle with diameter $d = 12$.
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$12\pi$. Apply $C = \pi d$ with $d = 12$.
$12\pi$. Apply $C = \pi d$ with $d = 12$.
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Find the area of a circle with radius $r = 6$.
Find the area of a circle with radius $r = 6$.
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$36\pi$. Apply $A = \pi r^2$ with $r = 6$.
$36\pi$. Apply $A = \pi r^2$ with $r = 6$.
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Find the radius $r$ of a circle with circumference $C = 14\pi$.
Find the radius $r$ of a circle with circumference $C = 14\pi$.
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$7$. Solve $14\pi = 2\pi r$ for $r$.
$7$. Solve $14\pi = 2\pi r$ for $r$.
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Find the diameter $d$ of a circle with circumference $C = 9\pi$.
Find the diameter $d$ of a circle with circumference $C = 9\pi$.
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$9$. Solve $9\pi = \pi d$ for $d$.
$9$. Solve $9\pi = \pi d$ for $d$.
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Find the radius $r$ of a circle with area $A = 49\pi$.
Find the radius $r$ of a circle with area $A = 49\pi$.
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$7$. Solve $49\pi = \pi r^2$ for $r$.
$7$. Solve $49\pi = \pi r^2$ for $r$.
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Find the area of a semicircle with radius $r = 8$.
Find the area of a semicircle with radius $r = 8$.
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$32\pi$. Apply semicircle area $\frac{1}{2}\pi r^2$ with $r = 8$.
$32\pi$. Apply semicircle area $\frac{1}{2}\pi r^2$ with $r = 8$.
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Find the area of a quarter circle with radius $r = 4$.
Find the area of a quarter circle with radius $r = 4$.
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$4\pi$. Apply quarter circle area $\frac{1}{4}\pi r^2$ with $r = 4$.
$4\pi$. Apply quarter circle area $\frac{1}{4}\pi r^2$ with $r = 4$.
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Find the volume of a cylinder with radius $r = 3$ and height $h = 10$.
Find the volume of a cylinder with radius $r = 3$ and height $h = 10$.
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$90\pi$. Apply $V = \pi r^2 h$ with $r = 3$ and $h = 10$.
$90\pi$. Apply $V = \pi r^2 h$ with $r = 3$ and $h = 10$.
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Find the volume of a cylinder with diameter $d = 8$ and height $h = 5$.
Find the volume of a cylinder with diameter $d = 8$ and height $h = 5$.
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$80\pi$. Use $r = 4$ from $d = 8$, then $V = \pi r^2 h$.
$80\pi$. Use $r = 4$ from $d = 8$, then $V = \pi r^2 h$.
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Find the height $h$ of a cylinder with volume $V = 50\pi$ and radius $r = 5$.
Find the height $h$ of a cylinder with volume $V = 50\pi$ and radius $r = 5$.
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$2$. Solve $50\pi = \pi(5^2)h$ for $h$.
$2$. Solve $50\pi = \pi(5^2)h$ for $h$.
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In the sector dissection argument, what is the height of the rearranged shape?
In the sector dissection argument, what is the height of the rearranged shape?
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$r$. Height equals the circle's radius in the rearranged shape.
$r$. Height equals the circle's radius in the rearranged shape.
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In the sector dissection argument, what is the approximate base length of the rearranged shape?
In the sector dissection argument, what is the approximate base length of the rearranged shape?
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Approximately $\pi r$. Base length is half the circumference $C = 2\pi r$.
Approximately $\pi r$. Base length is half the circumference $C = 2\pi r$.
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What sector-dissection arrangement leads to a rectangle-like shape for a circle of radius $r$?
What sector-dissection arrangement leads to a rectangle-like shape for a circle of radius $r$?
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Alternate sectors to form base $\pi r$ and height $r$. Rearranging pie-slice sectors forms an approximate rectangle.
Alternate sectors to form base $\pi r$ and height $r$. Rearranging pie-slice sectors forms an approximate rectangle.
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State the formula for the area of a circle with radius $r$.
State the formula for the area of a circle with radius $r$.
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$A = \pi r^2$. Area equals pi times the radius squared.
$A = \pi r^2$. Area equals pi times the radius squared.
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Identify the informal dissection idea that justifies $C = 2\pi r$ using a straightened circle.
Identify the informal dissection idea that justifies $C = 2\pi r$ using a straightened circle.
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Straighten the circle to a segment of length $2\pi r$. Imagine unrolling the circle's perimeter into a straight line.
Straighten the circle to a segment of length $2\pi r$. Imagine unrolling the circle's perimeter into a straight line.
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What is the definition of $\pi$ using circumference $C$ and diameter $d$?
What is the definition of $\pi$ using circumference $C$ and diameter $d$?
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$\pi = \frac{C}{d}$. Pi is defined as the ratio of circumference to diameter.
$\pi = \frac{C}{d}$. Pi is defined as the ratio of circumference to diameter.
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State the formula for the circumference of a circle in terms of radius $r$.
State the formula for the circumference of a circle in terms of radius $r$.
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$C = 2\pi r$. Circumference equals diameter times $\pi$, where diameter is $2r$.
$C = 2\pi r$. Circumference equals diameter times $\pi$, where diameter is $2r$.
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State the formula for the circumference of a circle in terms of diameter $d$.
State the formula for the circumference of a circle in terms of diameter $d$.
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$C = \pi d$. Circumference is the ratio $\pi$ times the diameter.
$C = \pi d$. Circumference is the ratio $\pi$ times the diameter.
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Which expression equals the area of a circle if $C = 2\pi r$ is known: $A = \frac{1}{2}Cr$?
Which expression equals the area of a circle if $C = 2\pi r$ is known: $A = \frac{1}{2}Cr$?
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$A = \frac{1}{2}Cr$. Alternative area formula using circumference and radius.
$A = \frac{1}{2}Cr$. Alternative area formula using circumference and radius.
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