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Geometry 9-12 / Week 12 / Thursday
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Week 12 · Modeling the Set and Opening Night

Thursday

Least fabric, most seats
// Every prop a shape, every plan a number
⏱ about 20 min

Thursday: Least Fabric, Most Seats

Wren pins three backdrop sketches to the corkboard. "Three ways to hang the fabric. Same stage, different amounts."

"A is one flat piece," Comet says. "B adds two wings. C is one wider piece. What do you notice?"

"A uses the least," Wren says, "if it still hides the back wall. That is the constraint."

Nova projects the floor plan: a 12 by 8 rectangle, an aisle down the middle. "Would you like a hint? Seats are 0.5 wide and 0.8 deep."

"Twenty-four across, minus two for the aisle. Ten rows," Comet says. "What can we make? Two hundred and twenty seats."

"Then the model," Wren says. "One centimeter for every twenty. Our designer, design under the constraints."

Design problems have constraints

A constraint is a limit the design must respect. The backdrop must hide the back wall. The seats must fit the floor with an aisle.

Geometry turns each option into a number. Then the crew picks the least material, or the most seats, that still meets the constraint.

The least fabric

All three options hide the back wall. Every length is the crew's own, in meters.

OptionPiecesFabric (square meters)
A: one flat piece6 by 424
B: flat piece plus two wings6 by 4 plus two of 1.5 by 436
C: one wider piece7 by 428

Option A uses the least fabric, 24 square meters. The wings in B add 12 square meters for no extra cover.

The most seats

The floor plan on the stage grid: a 12 by 8 rectangle of taped marks, in meters.

The floor is the rectangle (0, 0), (12, 0), (12, 8), (0, 8), area 96 square meters.

A seat with legroom is 0.5 m wide and 0.8 m deep. Across 12 m: 24 seats, minus 2 for the aisle, is 22.

Down 8 m: 10 rows. 22 × 10 = 220 seats, the most that fit with the aisle.

The model

The cardboard model uses 1 cm for every 20 cm on stage. A 30 cm model wall is 600 cm, or 6 m, on stage.

FABRIC UNDER CONSTRAINTS
  • Read the question.
  • Tap your answer.
A 6 by 4 m rectangle drawn on a gridOption A is one piece 6 m by 4 m. What is its area in square meters?
Option B adds two wings of 1.5 m by 4 m to option A. How many square meters of fabric is option B?
Which option uses the least fabric and still hides the back wall?
SEATS AND THE MODEL
  • Read the question.
  • Tap your answer.
A grid with a figure at (0, 0), (12, 0), (12, 8), (0, 8)The floor is the rectangle with corners (0, 0), (12, 0), (12, 8), (0, 8). What is its area in square meters?
Each row holds 22 seats after the aisle, and there are 10 rows. How many seats fit?
The model uses 1 cm for every 20 cm on stage. A wall is 30 cm on the model. How long is the real wall in centimeters?
A second theater builds the backdrop with every length doubled. The area was 24 square meters. What is the new area?
The floor is 12 m by 8 m. What is its area in square meters? Type the number.
WHY THIS EXERCISEThe floor area is the first constraint: every seat plan has to fit inside it.
Seats are 0.5 m wide. How many fit across a 12 m row before the aisle? Type the number.
Seats are 0.8 m deep. How many rows fit in 8 m? Type the number.
Try it
Sketch a room at home as a rectangle. Choose a chair size and an aisle. How many chairs fit?
Then try the chairs turned sideways. Does the count change?
Draw the 12 by 8 floor plan with the aisle down the middle. Sketch one row of seats and label the seat size.

Sharp design, designer. Tomorrow is Opening Night: the set under the lights, and the whole course in review.

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