Math III · G-MG.3

Using Geometric Methods to Solve Design Problems Under Constraints

Design problems force students to use geometry as decision-making: a shape must fit, cost, hold, support, or satisfy several limits at once.

Concept Geometry

Domain Modeling with Geometry

Read time 6 minutes

What this learning objective is really asking you to learn

This objective asks students to use geometric methods to solve design problems under constraints such as cost, space, or ratios. A design problem is not just “find the area” or “find the volume.” It usually asks for a choice that must satisfy several conditions.

For example:

Design a box that holds a certain volume but uses limited material.
Plan a garden with fixed fencing and required area.
Choose dimensions for a sign that fit a wall and stay within a budget.
Scale a model so it fits in a display case.
Design packaging that minimizes wasted space.
Arrange objects in a room while preserving walking space.
Determine whether a ramp meets length and slope constraints.

A constraint is a limit or requirement. Cost must be below a budget. Space must fit within dimensions. Volume must be at least a minimum. Material must not exceed a limit. Ratios may need to be maintained. Safety or accessibility rules may set maximum slopes or minimum clearances.

This objective asks students to use geometry to make decisions. They must choose variables, represent constraints, apply formulas, compare options, and interpret whether the design works. Sometimes there is one solution. Sometimes there are many viable solutions. Sometimes no solution meets all constraints. Sometimes the task is to choose the best among viable options.

The key is that geometry becomes a modeling and design tool. Shapes, measurements, area, volume, similarity, scale, and ratios are used to solve practical problems.

Why students should learn this math

Students should learn geometric design under constraints because almost every real design problem has limits. Architects deal with space, materials, cost, codes, and human use. Engineers deal with strength, weight, clearance, capacity, and safety. Product designers deal with packaging, volume, material, ergonomics, and manufacturing cost. Event planners deal with room layouts and capacity. Homeowners deal with budgets, dimensions, and materials. Cities deal with roads, parcels, parks, utilities, and density.

This objective teaches students that mathematics is often about tradeoffs. A larger container holds more but may cost more material. A more compact package saves shipping volume but may be harder to manufacture. A ramp with a gentler slope is safer but takes more space. A garden with maximum area under fixed fencing may require a particular shape. A design may be mathematically possible but too expensive.

Students also learn that constraints must be explicit. “Big enough” is vague. “At least 200 square feet” is mathematical. “Affordable” is vague. “Cost no more than $1,500” is mathematical. “Not too steep” is vague. “Slope no more than 1:12” is mathematical. Geometry helps turn vague design goals into measurable conditions.

This is a powerful life skill. Students learn to evaluate options rather than guess. They learn to justify design choices. They learn to notice when a proposed design violates a constraint. They learn that a solution is not just a number; it is a decision supported by reasoning.

The “why” is that design is geometry under pressure. Constraints make the geometry matter.

The historical machinery: geometry as design technology

Geometry has always been tied to design. Ancient builders used geometry to lay out buildings, fields, roads, temples, and monuments. Surveyors used geometry to divide land. Craftspeople used geometric patterns and proportions. Engineers used geometric reasoning for structures, machines, and tools.

Modern design still relies on geometry, now supported by digital tools. CAD systems use geometric models to test dimensions and constraints. Architects use floor plans, sections, and 3D models. Industrial designers optimize shapes for cost, strength, and appearance. Civil engineers design roads, bridges, drainage systems, and public spaces under strict constraints.

Optimization is the advanced mathematical version of design under constraints. Students are not doing full optimization theory here, but they are practicing the same mindset: define what is allowed, calculate consequences, and choose a viable or best option.

The historical lesson is that geometry is one of humanity's oldest design technologies.

Where this fits in the big map of mathematics

This objective follows geometric modeling and density. Students now combine those ideas with constraints and decisions.

It connects to systems of inequalities from algebra. Design constraints can be represented as equations or inequalities.

It connects to area, volume, surface area, scale, similarity, and density.

It connects to optimization. Some problems ask for maximum area, minimum cost, or best fit.

It connects to modeling. Assumptions, units, and feasibility matter.

It connects to engineering, architecture, product design, logistics, and construction.

The big-map role is geometric decision-making. Students learn to use measurement and shape to solve constrained real problems.

How to execute the skill technically

Use a design process:

Understand the design goal.
Identify constraints.
Define variables and units.
Draw a diagram.
Write geometric relationships.
Write cost, space, or ratio constraints.
Solve or compare options.
Check feasibility.
Interpret and justify the design.

Example: A rectangular garden must have area at least 120 square feet. Fencing costs $8 per foot, and the budget is $400. Can a 10 ft by 12 ft garden work?

Area:

$10 \cdot 12=120$, so it meets the area constraint.

Perimeter:

\[2(10)+2(12)=44\].

Cost:

\[44 \cdot 8=352\].

The cost is $352, under the $400 budget. The design is viable.

Could a 5 ft by 24 ft garden work?

Area:

\[5 \cdot 24=120\].

Perimeter:

\[2(5)+2(24)=58\].

Cost:

\[58 \cdot 8=464\].

Same area, but cost exceeds budget. Not viable.

This example shows that area alone is not enough. Shape affects perimeter and cost.

Worked example: packaging constraint

A company wants a rectangular box with volume at least 1,000 cubic centimeters. The base must be 10 cm by 10 cm because of the product footprint. What height is needed?

Volume:

\[V=lwh\].

With base 10 by 10:

\[V=100h\].

Require:

\[100h \ge 1000\].

\[h \ge 10\].

The box must be at least 10 cm tall.

If cardboard cost depends on surface area, the problem becomes richer. A closed box with dimensions 10 by 10 by h has surface area

\[SA = 2(10 \cdot 10)+4(10h)=200+40h\].

At $h=10$, surface area is 600 square centimeters. If cardboard costs $0.02 per square centimeter, material cost is $12.

This example connects volume constraint and cost constraint.

Worked example: scale-model display

A building is 80 meters long, 50 meters wide, and 30 meters tall. A scale model must fit inside a display case 40 cm long, 30 cm wide, and 20 cm tall. What scale factor from real building to model works?

Convert real dimensions to centimeters:

80 m = 8000 cm, 50 m = 5000 cm, 30 m = 3000 cm.

Scale constraints:

Length: $k \cdot 8000 \le 40$, so $k \le 1/200$.

Width: $k \cdot 5000 \le 30$, so $k \le 3/500 = 0.006$.

Height: $k \cdot 3000 \le 20$, so $k \le 1/150$.

The most restrictive is $1/200$. A scale of 1:200 fits all dimensions. This is geometric constraint reasoning with ratios.

More design example: ramp constraint

A wheelchair ramp must rise 2 feet. Suppose a design guideline requires at least 12 feet of ramp length for every 1 foot of rise. How long must the ramp be?

Required horizontal/ramp ratio:

12:1.

For rise 2 feet:

\[length \ge 12(2)=24\].

The ramp must be at least 24 feet long. But the design also needs available space. If the site has only 18 feet of straight-line space, the design is not viable unless the ramp turns or uses a different layout.

This example shows constraints interacting: accessibility slope and physical space.

Optimization flavor without calculus

Some design problems ask for the best shape under constraints. For example, among rectangles with fixed perimeter, the square has maximum area. Among boxes with fixed volume, dimensions affect surface area and cost. Students do not need calculus here, but they should see that geometry can compare alternatives.

Example: With 40 feet of fencing, compare a 5 by 15 rectangle and a 10 by 10 square. Both have perimeter 40. Areas are 75 and 100. The square gives more area. This is an accessible optimization insight.

Ratios in design

Design often includes ratios. A poster may need width-to-height ratio 2:3. A model may need scale 1:50. A mixture of spaces in a floor plan may require certain area ratios. If total area is 500 square feet and two zones must be in ratio 3:2, then the parts are 300 and 200 square feet.

Ratios help preserve shape, balance, and function. Students should learn to translate ratio constraints into equations.

Problem Library

Problems in the App From This Objective

147 problems across 12 archetypes in the app.

model dimensions and constraints.

12 problems Warmup Practice Mixed Review Assessment

Problem 1

Optimize area under constraint rectangle perimeter 40.

Problem 2

Optimize area under constraint three-sided pen with 60 ft fencing against wall.

Problem 3

Optimize area under constraint fixed perimeter P rectangle.

Problem 4

Optimize area under constraint rectangle perimeter 100.

Open in simulator

Problem 5

Optimize area under constraint three-sided pen with 100 ft fencing against wall.

Problem 6

Optimize area under constraint rectangular enclosure with 120 ft fencing against two perpendicular walls.

Problem 7

Optimize area under constraint rectangular pen with 120 ft fencing, divided into two sections by a fence parallel to one side.

Problem 8

Optimize area under constraint two adjacent rectangular pens sharing a side, total 200 ft fencing.

Problem 9

Optimize area under constraint rectangle with diagonal 10.

Problem 10

Optimize area under constraint rectangle inscribed in a right triangle with legs 6 and 8.

Problem 11

Optimize area under constraint three adjacent rectangular pens sharing sides, total 300 ft fencing.

Problem 12

Optimize area under constraint rectangular area where length plus twice width is 40.

model solids and constraints.

12 problems Warmup Practice Mixed Review Assessment

Problem 13

Optimize volume under material constraint open box from square sheet by cutting corner squares.

Problem 14

Optimize volume under material constraint closed rectangular box with fixed surface area.

Problem 15

Optimize volume under material constraint cylinder with fixed surface area.

Problem 16

Optimize volume under material constraint closed rectangular box with square base and fixed surface area.

Problem 17

Optimize volume under material constraint open rectangular box with square base and fixed surface area.

Problem 18

Optimize volume under material constraint cylinder with fixed volume and minimize surface area.

Problem 19

Optimize volume under material constraint closed rectangular box with fixed volume and minimize surface area.

Open in simulator

Problem 20

Optimize volume under material constraint open rectangular box with fixed volume and minimize surface area.

Problem 21

Optimize volume under material constraint cylindrical can where top/bottom material costs double, fixed total material cost.

Problem 22

Optimize volume under material constraint rectangular trough with fixed material for cross-section (open top), maximize cross-sectional area.

Problem 23

Optimize volume under material constraint open-top cylinder with fixed surface area.

Problem 24

Optimize volume under material constraint closed rectangular box, one face is a wall, fixed material for the other 5 faces.

compute material quantities and apply unit cost.

12 problems Warmup Practice Mixed Review Assessment

Problem 25

Minimize cost for design flooring 240 sq ft at 4 dollars/sq ft plus trim 60 ft at 2 dollars/ft.

Problem 26

Minimize cost for design paint 900 sq ft, 350 sq ft/gallon, 32 dollars/gallon.

Problem 27

Minimize cost for design box material areas A1,A2 with unit costs c1,c2.

Problem 28

Minimize cost for design fencing a rectangular garden 20 ft by 30 ft at 5 dollars/ft.

Problem 29

Minimize cost for design paint walls (1200 sq ft) and ceiling (300 sq ft), 400 sq ft/gallon, 25 dollars/gallon.

Problem 30

Minimize cost for design tile bathroom floor 80 sq ft at 3 dollars/sq ft and shower wall 60 sq ft at 5 dollars/sq ft.

Problem 31

Minimize cost for design carpet a circular room with radius 10 ft at 6 dollars/sq ft.

Problem 32

Minimize cost for design roofing a shed 10 ft by 12 ft, shingles cost 75 dollars per 100 sq ft bundle.

Problem 33

Minimize cost for design packaging material for a box 2 ft x 3 ft x 4 ft, material costs 0.5 dollars/sq ft.

Problem 34

Minimize cost for design concrete for a patio 10 ft by 15 ft, 0.5 ft deep, concrete costs 100 dollars/cubic yard.

Open in simulator

Problem 35

Minimize cost for design wallpapering a wall 10 ft by 8 ft with a 3 ft by 4 ft window, wallpaper costs 2 dollars/sq ft.

Problem 36

Minimize cost for design flooring for living room 300 sq ft at 5 dollars/sq ft and kitchen 150 sq ft at 7 dollars/sq ft.

apply proportional length/area/volume relationships.

12 problems Warmup Practice Mixed Review Assessment

Problem 37

Use ratios to scale design length scale factor 3.

Problem 38

Use ratios to scale design model 1:20.

Problem 39

Use ratios to scale design double all lengths.

Problem 40

Use ratios to scale design length scale factor 4.

Problem 41

Use ratios to scale design halve all lengths.

Problem 42

Use ratios to scale design model 1:10.

Problem 43

Use ratios to scale design triple all lengths.

Problem 44

Use ratios to scale design length scale factor 0.5.

Problem 45

Use ratios to scale design blueprint scale 1:60.

Problem 46

Use ratios to scale design lengths are reduced to 1/5.

Problem 47

Use ratios to scale design areas scale by 25.

Open in simulator

Problem 48

Use ratios to scale design volumes scale by 8.

compare object dimensions and available space.

12 problems Warmup Practice Mixed Review Assessment

Problem 49

Solve packing or fitting problem boxes 2x3x4 in container 10x9x8 aligned.

Problem 50

Solve packing or fitting problem circle diameter 4 inside square side 5.

Problem 51

Solve packing or fitting problem cylinder height 12 in shelf height 10 in upright.

Problem 52

Solve packing or fitting problem boxes 1x2x3 in container 5x4x6 aligned.

Problem 53

Solve packing or fitting problem cylinder radius 3 height 8 in box 6x6x10 upright.

Problem 54

Solve packing or fitting problem cylinder radius 4 height 15 in box 8x8x12 upright.

Problem 55

Solve packing or fitting problem sphere radius 5 in box 10x10x10.

Problem 56

Solve packing or fitting problem circles diameter 2 in rectangle 5x3.

Open in simulator

Problem 57

Solve packing or fitting problem rectangle 3x5 in rectangle 6x4.

Problem 58

Solve packing or fitting problem boxes 3x3x3 in container 5x5x2 aligned.

Problem 59

Solve packing or fitting problem square side 7 in circle diameter 10.

Problem 60

Solve packing or fitting problem cubes side 2 in a line of length 10.

test area, volume, cost, and space limits.

12 problems Warmup Practice Mixed Review Assessment

Problem 61

Choose feasible design from candidates design A area 90 cost 120; limit area at least 80 and cost at most 150.

Problem 62

Choose feasible design from candidates design B volume 50 but required capacity 60.

Problem 63

Choose feasible design from candidates multiple candidates with area, volume, cost limits.

Problem 64

Choose feasible design from candidates design C area 100 volume 200 cost 500; limits area at most 90, volume at least 150, cost at most 600.

Problem 65

Choose feasible design from candidates design X area 70 cost 100; design Y area 85 cost 120; limits area at least 80, cost at most 130.

Problem 66

Choose feasible design from candidates design P volume 40 height 5; design Q volume 60 height 6; limits volume at least 70, height at most 5.5.

Problem 67

Choose feasible design from candidates design D length 10 width 5 height 3; limits length at most 12, width at least 4, height at most 3.

Problem 68

Choose feasible design from candidates design R capacity 100 price 200; design S capacity 90 price 150; design T capacity 98 price 170; limits capacity at least 95, price at most 180.

Problem 69

Choose feasible design from candidates design E power 500 efficiency 0.8; design F power 490 efficiency 0.9; limits power at least 480, efficiency at least 0.85.

Problem 70

Choose feasible design from candidates design G temperature 25 pressure 10; limits temperature between 20 and 30, pressure at most 12.

Open in simulator

Problem 71

Choose feasible design from candidates model 1 size 100 cost 50; model 2 size 120 cost 60; model 3 size 115 cost 52; limits size at least 110, cost at most 55.

Problem 72

Choose feasible design from candidates option Alpha weight 15kg speed 50km/h; option Beta weight 12kg speed 60km/h; limits weight at most 14kg, speed at least 55km/h.

decompose and combine measurements.

12 problems Warmup Practice Mixed Review Assessment

Problem 73

Model design with composite shapes garden rectangle plus semicircle end.

Open in simulator

Problem 74

Model design with composite shapes floor plan two rectangles.

Problem 75

Model design with composite shapes container cylinder plus cone top.

Problem 76

Model design with composite shapes L-shaped room.

Problem 77

Model design with composite shapes square plate with a circular hole.

Problem 78

Model design with composite shapes building rectangular prism with triangular prism roof.

Problem 79

Model design with composite shapes running track with straight sections and semicircular ends.

Problem 80

Model design with composite shapes silo cylinder with hemispherical top.

Problem 81

Model design with composite shapes overlapping circular regions.

Problem 82

Model design with composite shapes capsule cylinder with two hemispherical ends.

Problem 83

Model design with composite shapes composite shape rectangle and trapezoid.

Problem 84

Model design with composite shapes rectangular block with a cylindrical hole.

combine geometry and density.

12 problems Warmup Practice Mixed Review Assessment

Problem 85

Solve design problem involving density or capacity tank volume 4 m^3, water density 1000 kg/m^3.

Problem 86

Solve design problem involving density or capacity floor load limit 60 lb/sq ft, platform area 20 sq ft.

Problem 87

Solve design problem involving density or capacity soil density 80 lb/ft^3, planter volume 15 ft^3.

Problem 88

Solve design problem involving density or capacity concrete density 2400 kg/m^3, slab volume 0.5 m^3.

Problem 89

Solve design problem involving density or capacity shelf load capacity 50 kg/m^2, shelf area 1.5 m^2.

Problem 90

Solve design problem involving density or capacity crate capacity 12 items, 25 crates.

Problem 91

Solve design problem involving density or capacity wood density 40 lb/ft^3, beam volume 2.5 ft^3.

Problem 92

Solve design problem involving density or capacity paint coverage 350 sq ft/gallon, wall area 700 sq ft.

Problem 93

Solve design problem involving density or capacity pool volume 50 m^3, water capacity 1000 liters/m^3.

Open in simulator

Problem 94

Solve design problem involving density or capacity pipe length 15 meters, linear density 5 kg/meter.

Problem 95

Solve design problem involving density or capacity flooring cost $25/sq ft, room area 150 sq ft.

Problem 96

Solve design problem involving density or capacity desired liquid mass 600 kg, liquid density 800 kg/m^3.

solve angled measurements.

15 problems Warmup Practice Mixed Review Assessment

Problem 97

Use trigonometry in design constraint ramp rises 3 ft at angle 6 degrees.

Problem 98

Use trigonometry in design constraint brace length 10 ft at 30 degrees to floor.

Problem 99

Use trigonometry in design constraint roof rise 4 ft over run 12 ft.

Problem 100

Use trigonometry in design constraint ramp length 15 ft at angle 10 degrees to ground.

Problem 101

Use trigonometry in design constraint roof rise 6 ft with angle 25 degrees.

Problem 102

Use trigonometry in design constraint brace 8 ft long supports a post 5 ft from its base.

Problem 103

Use trigonometry in design constraint ramp rises 4 ft over a horizontal distance of 20 ft.

Problem 104

Use trigonometry in design constraint roof rafter length 20 ft at 35 degrees to horizontal.

Problem 105

Use trigonometry in design constraint brace 12 ft long makes an angle of 40 degrees with the floor.

Problem 106

Use trigonometry in design constraint ramp has a run of 18 ft and an angle of 8 degrees.

Problem 107

Use trigonometry in design constraint roof rafter length 25 ft at 20 degrees to horizontal.

Problem 108

Use trigonometry in design constraint brace 10 ft long makes 50 degrees with the vertical wall.

Problem 109

Use trigonometry in design constraint ramp has a rise of 5 ft and a run of 20 ft.

Open in simulator

Problem 110

Use trigonometry in design constraint roof rafter length 18 ft and rise 7 ft.

Problem 111

Use trigonometry in design constraint brace needs to support a point 7 ft up a wall, and extend 4 ft from the wall.

evaluate cost, space, material, and performance.

12 problems Warmup Practice Mixed Review Assessment

Problem 112

Compare design tradeoffs for cheaper material has shorter lifespan.

Open in simulator

Problem 113

Compare design tradeoffs for larger tank costs more but reduces refill frequency.

Problem 114

Compare design tradeoffs for lighter material reduces load but increases price.

Problem 115

Compare design tradeoffs for a faster processor consumes more power.

Problem 116

Compare design tradeoffs for a compact design limits battery size.

Problem 117

Compare design tradeoffs for a more durable coating increases application time.

Problem 118

Compare design tradeoffs for a less expensive cooling system requires more space.

Problem 119

Compare design tradeoffs for a higher resolution display increases power consumption.

Problem 120

Compare design tradeoffs for using recycled materials might reduce structural integrity.

Problem 121

Compare design tradeoffs for a smaller factory floor reduces rent but limits output.

Problem 122

Compare design tradeoffs for a high-precision sensor is more expensive and fragile.

Problem 123

Compare design tradeoffs for thicker insulation improves thermal efficiency but reduces usable space.

identify simplifications and measurement limits.

12 problems Warmup Practice Mixed Review Assessment

Problem 124

State assumptions in geometric design model paint estimate for rough wall.

Problem 125

State assumptions in geometric design model box modeled as rectangular prism.

Problem 126

State assumptions in geometric design model tank modeled as cylinder.

Problem 127

State assumptions in geometric design model surface area of a lake for environmental assessment.

Open in simulator

Problem 128

State assumptions in geometric design model timber volume of a log modeled as a cylinder.

Problem 129

State assumptions in geometric design model footprint area of an L-shaped building.

Problem 130

State assumptions in geometric design model volume of a grain silo modeled as a cylinder with a conical base.

Problem 131

State assumptions in geometric design model radius of curvature for a highway bend.

Problem 132

State assumptions in geometric design model surface area of a geodesic dome modeled as a hemisphere.

Problem 133

State assumptions in geometric design model volume of soil for a trench excavation.

Problem 134

State assumptions in geometric design model volume of water in a small pond for irrigation.

Problem 135

State assumptions in geometric design model length of fencing required for an irregular property boundary.

catch missing constraint, wrong formula, unit, or feasibility mistake.

12 problems Warmup Practice Mixed Review Assessment

Problem 136

Correct geometric design-modeling error ignored budget constraint while maximizing area.

Problem 137

Correct geometric design-modeling error used circumference instead of area for flooring.

Problem 138

Correct geometric design-modeling error fit decision based only on volume.

Problem 139

Correct geometric design-modeling error forgot unit conversion from inches to feet before costing.

Problem 140

Correct geometric design-modeling error calculated fence length using the area of the garden.

Problem 141

Correct geometric design-modeling error estimated paint needed for a box using its volume.

Problem 142

Correct geometric design-modeling error mixed cubic feet and cubic inches directly for total volume.

Open in simulator

Problem 143

Correct geometric design-modeling error designed a rectangular pool without considering depth limit.

Problem 144

Correct geometric design-modeling error used the area formula for a square to calculate the area of a circle.

Problem 145

Correct geometric design-modeling error derived a negative length for a side of a building.

Problem 146

Correct geometric design-modeling error applied a 1:10 scale factor to all dimensions but forgot to square it for area.

Problem 147

Correct geometric design-modeling error maximized the perimeter instead of the area for a given amount of material.