Using Geometry to Solve Design Problems - Geometry
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What is the arc length of a circle sector with radius $r$ and central angle $ heta$ (radians)?
What is the arc length of a circle sector with radius $r$ and central angle $ heta$ (radians)?
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$s = r heta$. Arc length: radius times central angle in radians.
$s = r heta$. Arc length: radius times central angle in radians.
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What is the relationship between scale factor $k$ and volume scaling for similar solids?
What is the relationship between scale factor $k$ and volume scaling for similar solids?
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Volumes scale by $k^3$. Volume scaling follows the cube of linear scale factor.
Volumes scale by $k^3$. Volume scaling follows the cube of linear scale factor.
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What is the midpoint formula for endpoints $(x_1,y_1)$ and $(x_2,y_2)$?
What is the midpoint formula for endpoints $(x_1,y_1)$ and $(x_2,y_2)$?
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$M = (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$. Midpoint: average of corresponding coordinates.
$M = (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$. Midpoint: average of corresponding coordinates.
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Find the missing width if a rectangle has height $10$ and aspect ratio $3:2$ (width:height).
Find the missing width if a rectangle has height $10$ and aspect ratio $3:2$ (width:height).
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Width $= 15$. From ratio $3:2 = w:10$, solve $w = 15$.
Width $= 15$. From ratio $3:2 = w:10$, solve $w = 15$.
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Identify the area of a $12$ by $7$ rectangular patio for paving cost calculations.
Identify the area of a $12$ by $7$ rectangular patio for paving cost calculations.
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Area $= 84$. Area: $A = 12 \times 7 = 84$.
Area $= 84$. Area: $A = 12 \times 7 = 84$.
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Identify the radius needed for a circular table with circumference $62.8$ using $\text{π} \approx 3.14$.
Identify the radius needed for a circular table with circumference $62.8$ using $\text{π} \approx 3.14$.
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Radius $= 10$. From $C = 2\pi r = 62.8$, solve $r = 10$.
Radius $= 10$. From $C = 2\pi r = 62.8$, solve $r = 10$.
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Find the surface area of a sphere with radius $2$ for coating material estimation.
Find the surface area of a sphere with radius $2$ for coating material estimation.
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Surface area $= 16\text{π}$. Sphere surface area: $SA = 4\pi r^2 = 16\pi$.
Surface area $= 16\text{π}$. Sphere surface area: $SA = 4\pi r^2 = 16\pi$.
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Which value of $m$ makes a line perpendicular to a line with slope $\frac{3}{4}$?
Which value of $m$ makes a line perpendicular to a line with slope $\frac{3}{4}$?
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$m = -\frac{4}{3}$. Perpendicular slope is negative reciprocal: $-\frac{1}{\frac{3}{4}} = -\frac{4}{3}$.
$m = -\frac{4}{3}$. Perpendicular slope is negative reciprocal: $-\frac{1}{\frac{3}{4}} = -\frac{4}{3}$.
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Identify the midpoint of segment endpoints $(2,8)$ and $(10,4)$ for centering a support.
Identify the midpoint of segment endpoints $(2,8)$ and $(10,4)$ for centering a support.
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Midpoint $= (6,6)$. Midpoint: $(\frac{2+10}{2}, \frac{8+4}{2}) = (6,6)$.
Midpoint $= (6,6)$. Midpoint: $(\frac{2+10}{2}, \frac{8+4}{2}) = (6,6)$.
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Find the height if a rectangle has width $24$ and aspect ratio $4:3$ (width:height).
Find the height if a rectangle has width $24$ and aspect ratio $4:3$ (width:height).
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Height $= 18$. From ratio $4:3 = 24:h$, solve $h = 18$.
Height $= 18$. From ratio $4:3 = 24:h$, solve $h = 18$.
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Find the missing width if a rectangle has height $10$ and aspect ratio $3:2$ (width:height).
Find the missing width if a rectangle has height $10$ and aspect ratio $3:2$ (width:height).
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Width $= 15$. From ratio $3:2 = w:10$, solve $w = 15$.
Width $= 15$. From ratio $3:2 = w:10$, solve $w = 15$.
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What is the aspect ratio definition for a rectangle with width $W$ and height $H$?
What is the aspect ratio definition for a rectangle with width $W$ and height $H$?
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Aspect ratio $= W:H$ (or $\frac{W}{H}$). Ratio comparing width to height proportions.
Aspect ratio $= W:H$ (or $\frac{W}{H}$). Ratio comparing width to height proportions.
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What is the golden ratio as an exact expression?
What is the golden ratio as an exact expression?
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$\varphi = \frac{1+\text{√}5}{2}$. Mathematical constant approximately $1.618$.
$\varphi = \frac{1+\text{√}5}{2}$. Mathematical constant approximately $1.618$.
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What is a typical cost model for paving: cost per unit $c$ and area $A$?
What is a typical cost model for paving: cost per unit $c$ and area $A$?
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Total cost $= cA$. Linear cost model based on area for materials.
Total cost $= cA$. Linear cost model based on area for materials.
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What is a typical cost model for fencing: cost per unit $c$ and perimeter $P$?
What is a typical cost model for fencing: cost per unit $c$ and perimeter $P$?
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Total cost $= cP$. Linear cost model based on perimeter for fencing.
Total cost $= cP$. Linear cost model based on perimeter for fencing.
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What is the slope formula between points $(x_1,y_1)$ and $(x_2,y_2)$?
What is the slope formula between points $(x_1,y_1)$ and $(x_2,y_2)$?
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$m = \frac{y_2-y_1}{x_2-x_1}$. Slope: rise over run between two points.
$m = \frac{y_2-y_1}{x_2-x_1}$. Slope: rise over run between two points.
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What is the relationship between scale factor $k$ and volume scaling for similar solids?
What is the relationship between scale factor $k$ and volume scaling for similar solids?
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Volumes scale by $k^3$. Volume scaling follows the cube of linear scale factor.
Volumes scale by $k^3$. Volume scaling follows the cube of linear scale factor.
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What is the relationship between scale factor $k$ and area scaling for similar figures?
What is the relationship between scale factor $k$ and area scaling for similar figures?
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Areas scale by $k^2$. Area scaling follows the square of linear scale factor.
Areas scale by $k^2$. Area scaling follows the square of linear scale factor.
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What is the volume of a sphere with radius $r$?
What is the volume of a sphere with radius $r$?
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$V = \frac{4}{3} \pi r^3$. Sphere volume: four-thirds $\pi$ times radius cubed.
$V = \frac{4}{3} \pi r^3$. Sphere volume: four-thirds $\pi$ times radius cubed.
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What is the circumference of a circle with radius $r$?
What is the circumference of a circle with radius $r$?
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$C = 2\text{π}r$. Circle circumference: $2\pi$ times radius.
$C = 2\text{π}r$. Circle circumference: $2\pi$ times radius.
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What is the volume of a cone with radius $r$ and height $h$?
What is the volume of a cone with radius $r$ and height $h$?
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$V = \frac{1}{3} \pi r^2 h$. Cone volume: one-third of cylinder volume.
$V = \frac{1}{3} \pi r^2 h$. Cone volume: one-third of cylinder volume.
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What is the area of a circle with radius $r$?
What is the area of a circle with radius $r$?
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$A = \pi r^2$. Circle area: $\pi$ times radius squared.
$A = \pi r^2$. Circle area: $\pi$ times radius squared.
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What is the lateral area of a cylinder with radius $r$ and height $h$?
What is the lateral area of a cylinder with radius $r$ and height $h$?
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$LA = 2\pi rh$. Lateral area: circumference times height (curved surface only).
$LA = 2\pi rh$. Lateral area: circumference times height (curved surface only).
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What is the scale factor from a drawing to the real object if $1$ unit represents $k$ units?
What is the scale factor from a drawing to the real object if $1$ unit represents $k$ units?
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Scale factor is $k$ (multiply drawing lengths by $k$). Conversion factor from drawing to real dimensions.
Scale factor is $k$ (multiply drawing lengths by $k$). Conversion factor from drawing to real dimensions.
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What does it mean for two shapes to be similar in a scale-model design?
What does it mean for two shapes to be similar in a scale-model design?
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Corresponding angles equal and side lengths proportional. Definition of similar figures for scaling calculations.
Corresponding angles equal and side lengths proportional. Definition of similar figures for scaling calculations.
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What is the objective function in a geometric design optimization problem?
What is the objective function in a geometric design optimization problem?
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The quantity to minimize or maximize (often cost, area, or volume). The goal to optimize in design problems.
The quantity to minimize or maximize (often cost, area, or volume). The goal to optimize in design problems.
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What is the definition of a geometric constraint in a design problem?
What is the definition of a geometric constraint in a design problem?
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A required condition on lengths, angles, area, or volume. Physical limitations that must be satisfied in design.
A required condition on lengths, angles, area, or volume. Physical limitations that must be satisfied in design.
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What is the area of a sector with radius $r$ and central angle $ heta$ (radians)?
What is the area of a sector with radius $r$ and central angle $ heta$ (radians)?
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$A = rac{1}{2}r^2 heta$. Sector area: half radius squared times angle in radians.
$A = rac{1}{2}r^2 heta$. Sector area: half radius squared times angle in radians.
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What is the arc length of a circle sector with radius $r$ and central angle $ heta$ (radians)?
What is the arc length of a circle sector with radius $r$ and central angle $ heta$ (radians)?
Tap to reveal answer
$s = r heta$. Arc length: radius times central angle in radians.
$s = r heta$. Arc length: radius times central angle in radians.
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What is the area of a regular polygon with perimeter $P$ and apothem $a$?
What is the area of a regular polygon with perimeter $P$ and apothem $a$?
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$A = rac{1}{2}aP$. Regular polygon area: half apothem times perimeter.
$A = rac{1}{2}aP$. Regular polygon area: half apothem times perimeter.
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