Same rope, different land
A farmer has 24 m of rope. He wants a goat pen, a fenced place for goats. Will every pen shape give the same space inside?
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The rule
No. The same rope gives different areas. The area is the space inside the pen. The rope's length is the perimeter.
Three pens, all made with the same 24 m of rope:
| Pen | Perimeter | Area |
|---|---|---|
| 6 m square | 6+6+6+6 = 24 m | 6 × 6 = 36 m² |
| 8 m by 4 m | 8+4+8+4 = 24 m | 8 × 4 = 32 m² |
| 10 m by 2 m | 10+2+10+2 = 24 m | 10 × 2 = 20 m² |
Same rope every time. The area gets smaller down the table. The pen closest to a square has the most area.
A shorter rope can still give more area. A 5 m square needs only 20 m of rope. 5 + 5 + 5 + 5 = 20 m. The square's area is 5 × 5 = 25 m². The 10 m by 2 m pen needs 24 m of rope. Its area is only 20 m².
Slide the length in the lab. The rope stays 24 m.