A circle, a square and an equilateral triangle have the same area. The correct increasing order of the perimeters will be

### Concept & Isoperimetric Property (Inverse) The Isoperimetric principle establishes that a circle is the most efficient shape for enclosing area. This means for a fixed area, a circle will require the least amount of perimeter. Conversely, shapes with fewer sides are less efficient and require a larger perimeter to enclose that same area. ### Step-by-Step Solution 1. **Understand Efficiency:** Efficiency here means getting the most area for the least perimeter. - Most efficient: Circle - Moderately efficient: Square (4 sides) - Least efficient: Equilateral Triangle (3 sides) 2. **Order by Perimeter (Least to Greatest):** Since they all have the same area, the most efficient shape will have the smallest perimeter. - Smallest perimeter: Circle - Next largest: Square - Largest perimeter: Triangle 3. **Format the Output:** The increasing order (smallest to largest) is circle, square, triangle. ### Exam Strategy & Shortcut This is the inverse of the standard Isoperimetric rule. If equal perimeters mean $Area_{circle} > Area_{square} > Area_{triangle}$, then for equal areas, the perimeters must compensate in reverse: $Perimeter_{circle} < Perimeter_{square} < Perimeter_{triangle}$. Just remember: the circle always "wins" by doing the most with the least. ### Common Pitfall Confusing this question with the "equal perimeter" variation. Students often skim the question, assume it's asking for area based on equal perimeters, and pick the exact opposite order. Always underline whether "Area" or "Perimeter" is being held constant. ### Final Answer Therefore, the correct answer is **circle, square, triangle**.