Modeling with Geometric Shapes and Properties - Geometry
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What is the formula for the surface area of a rectangular prism with dimensions $l,w,h$?
What is the formula for the surface area of a rectangular prism with dimensions $l,w,h$?
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$S=2(lw+lh+wh)$. Sum of areas of all six rectangular faces.
$S=2(lw+lh+wh)$. Sum of areas of all six rectangular faces.
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What is the lateral surface area of a cone with $r=3$ and $\ell=8$?
What is the lateral surface area of a cone with $r=3$ and $\ell=8$?
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$L=24\pi$. Using $L = \pi r\ell = \pi(3)(8) = 24\pi$.
$L=24\pi$. Using $L = \pi r\ell = \pi(3)(8) = 24\pi$.
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Which 3D shape best models a soccer ball or marble?
Which 3D shape best models a soccer ball or marble?
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Sphere. Perfectly round surface equidistant from center matches spherical objects.
Sphere. Perfectly round surface equidistant from center matches spherical objects.
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Which 3D shape best models a shoebox or brick?
Which 3D shape best models a shoebox or brick?
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Rectangular prism. Six rectangular faces forming a box shape matches these objects.
Rectangular prism. Six rectangular faces forming a box shape matches these objects.
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Which 3D shape best models a soccer ball or marble?
Which 3D shape best models a soccer ball or marble?
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Sphere. Perfectly round surface equidistant from center matches spherical objects.
Sphere. Perfectly round surface equidistant from center matches spherical objects.
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Which 3D shape best models a party hat or ice cream cone?
Which 3D shape best models a party hat or ice cream cone?
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Right circular cone. Has circular base tapering to a point, matching the conical shape.
Right circular cone. Has circular base tapering to a point, matching the conical shape.
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What is the formula for the lateral surface area of a right circular cone?
What is the formula for the lateral surface area of a right circular cone?
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$L=\pi r\ell$. Curved surface area using radius and slant height.
$L=\pi r\ell$. Curved surface area using radius and slant height.
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What is the main purpose of geometric modeling in CCSS.G-MG.1?
What is the main purpose of geometric modeling in CCSS.G-MG.1?
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Represent real objects with ideal geometric figures to estimate measures. Approximates real-world shapes with ideal geometric forms for calculations.
Represent real objects with ideal geometric figures to estimate measures. Approximates real-world shapes with ideal geometric forms for calculations.
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Which measurement is typically used to find cylinder radius when modeling a torso as a cylinder?
Which measurement is typically used to find cylinder radius when modeling a torso as a cylinder?
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Use waist (or chest) circumference to get $r=\frac{C}{2\pi}$. Body circumference divided by $2\pi$ gives cylinder radius.
Use waist (or chest) circumference to get $r=\frac{C}{2\pi}$. Body circumference divided by $2\pi$ gives cylinder radius.
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Which measurement is typically used as the cylinder height when modeling a human torso?
Which measurement is typically used as the cylinder height when modeling a human torso?
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Torso length (vertical height of the modeled region). Vertical distance corresponding to cylinder height dimension.
Torso length (vertical height of the modeled region). Vertical distance corresponding to cylinder height dimension.
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What is the formula for the area of a circle with radius $r$?
What is the formula for the area of a circle with radius $r$?
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$A=\pi r^2$. Interior space of circle using radius squared and $\pi$.
$A=\pi r^2$. Interior space of circle using radius squared and $\pi$.
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What is the formula for the circumference of a circle with radius $r$?
What is the formula for the circumference of a circle with radius $r$?
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$C=2\pi r$. Distance around circle using radius and $\pi$.
$C=2\pi r$. Distance around circle using radius and $\pi$.
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What is the formula for the volume of a prism with base area $B$ and height $h$?
What is the formula for the volume of a prism with base area $B$ and height $h$?
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$V=Bh$. Base area times height for any prism with parallel bases.
$V=Bh$. Base area times height for any prism with parallel bases.
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A torso has circumference $C=80\pi$ and height $h=50$; what is cylinder volume?
A torso has circumference $C=80\pi$ and height $h=50$; what is cylinder volume?
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$V=80000\pi$. Using $r = \frac{80\pi}{2\pi} = 40$, so $V = \pi(1600)(50) = 80000\pi$.
$V=80000\pi$. Using $r = \frac{80\pi}{2\pi} = 40$, so $V = \pi(1600)(50) = 80000\pi$.
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What is a common limitation when modeling a real object as a perfect cylinder?
What is a common limitation when modeling a real object as a perfect cylinder?
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Real cross-sections and sides are not perfectly circular or straight. Real objects have irregularities not captured by perfect geometry.
Real cross-sections and sides are not perfectly circular or straight. Real objects have irregularities not captured by perfect geometry.
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What is the lateral surface area of a cylinder with $r=4$ and $h=10$?
What is the lateral surface area of a cylinder with $r=4$ and $h=10$?
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$L=80\pi$. Using $L = 2\pi rh = 2\pi(4)(10) = 80\pi$.
$L=80\pi$. Using $L = 2\pi rh = 2\pi(4)(10) = 80\pi$.
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What is the volume of a cylinder with $r=3$ and $h=5$?
What is the volume of a cylinder with $r=3$ and $h=5$?
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$V=45\pi$. Using $V = \pi r^2 h = \pi(3^2)(5) = 45\pi$.
$V=45\pi$. Using $V = \pi r^2 h = \pi(3^2)(5) = 45\pi$.
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Which measurement is most appropriate to estimate liquid capacity of a cylindrical tank?
Which measurement is most appropriate to estimate liquid capacity of a cylindrical tank?
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Volume $V=\pi r^2h$. Interior space determines how much liquid fits inside.
Volume $V=\pi r^2h$. Interior space determines how much liquid fits inside.
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Which measurement is most appropriate to estimate paint needed for a cylindrical can (ignore top and bottom)?
Which measurement is most appropriate to estimate paint needed for a cylindrical can (ignore top and bottom)?
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Lateral surface area $L=2\pi rh$. Curved side surface area determines paint coverage needed.
Lateral surface area $L=2\pi rh$. Curved side surface area determines paint coverage needed.
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Identify the best model for a tent with two triangular ends and rectangular sides.
Identify the best model for a tent with two triangular ends and rectangular sides.
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Triangular prism. Triangular cross-section extended lengthwise creates tent shape.
Triangular prism. Triangular cross-section extended lengthwise creates tent shape.
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Which quantity is the lateral area of a cylinder interpreted as a rectangle when unrolled?
Which quantity is the lateral area of a cylinder interpreted as a rectangle when unrolled?
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Rectangle with area $2\pi rh$. Circumference times height when curved surface is flattened.
Rectangle with area $2\pi rh$. Circumference times height when curved surface is flattened.
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Identify the best model for a drinking straw when thickness is negligible compared to length.
Identify the best model for a drinking straw when thickness is negligible compared to length.
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Cylinder (often treated as a hollow cylinder if thickness matters). Long thin circular object approximated as hollow or solid cylinder.
Cylinder (often treated as a hollow cylinder if thickness matters). Long thin circular object approximated as hollow or solid cylinder.
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Identify the best model for a stack of coins with flat circular faces.
Identify the best model for a stack of coins with flat circular faces.
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Cylinder. Multiple circular discs stacked create cylindrical shape.
Cylinder. Multiple circular discs stacked create cylindrical shape.
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A rectangular prism has $l=5$, $w=2$, $h=3$; what is its volume?
A rectangular prism has $l=5$, $w=2$, $h=3$; what is its volume?
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$V=30$. Using $V = lwh = 5(2)(3) = 30$.
$V=30$. Using $V = lwh = 5(2)(3) = 30$.
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A cylindrical can has $r=3$ and $h=12$; what is lateral area for the label?
A cylindrical can has $r=3$ and $h=12$; what is lateral area for the label?
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$L=72\pi$. Using $L = 2\pi rh = 2\pi(3)(12) = 72\pi$.
$L=72\pi$. Using $L = 2\pi rh = 2\pi(3)(12) = 72\pi$.
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A tree trunk has diameter $d=14$ and height $h=20$; what is cylinder volume?
A tree trunk has diameter $d=14$ and height $h=20$; what is cylinder volume?
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$V=980\pi$. Using $r = 7$, so $V = \pi(49)(20) = 980\pi$.
$V=980\pi$. Using $r = 7$, so $V = \pi(49)(20) = 980\pi$.
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What is the diameter of a circle with circumference $C=12\pi$?
What is the diameter of a circle with circumference $C=12\pi$?
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$d=12$. Using $d = \frac{C}{\pi} = \frac{12\pi}{\pi} = 12$.
$d=12$. Using $d = \frac{C}{\pi} = \frac{12\pi}{\pi} = 12$.
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What is the radius of a circular cross-section with circumference $C=18\pi$?
What is the radius of a circular cross-section with circumference $C=18\pi$?
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$r=9$. Using $r = \frac{C}{2\pi} = \frac{18\pi}{2\pi} = 9$.
$r=9$. Using $r = \frac{C}{2\pi} = \frac{18\pi}{2\pi} = 9$.
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What is the volume of a sphere with radius $r=2$?
What is the volume of a sphere with radius $r=2$?
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$V=\frac{32}{3}\pi$. Using $V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi(8) = \frac{32}{3}\pi$.
$V=\frac{32}{3}\pi$. Using $V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi(8) = \frac{32}{3}\pi$.
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What is the surface area of a sphere with radius $r=3$?
What is the surface area of a sphere with radius $r=3$?
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$S=36\pi$. Using $S = 4\pi r^2 = 4\pi(9) = 36\pi$.
$S=36\pi$. Using $S = 4\pi r^2 = 4\pi(9) = 36\pi$.
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