A circle, square, equilateral triangle and regular pentagon have the same areas. Arrange the following in ascending order based on their perimeters.

(A) Circle

(B) Square

(C) Equilateral triangle

(D) Regular pentagon

Don't worry - this looks scary but it's actually pretty straightforward!

More sides = lower perimeter.

Hence, circle has the lowest perimeter (it's made of infinite sides per se).

(A) Circle = Infinite sides.

(D) Pentagon = 5 sides.

(B) Square = 4 sides.

(C) Triangle = 3 sides.

---

Key insight: The circle is nature's most "efficient" shape - it packs the most area using the least perimeter. Other shapes "waste" more boundary for the same area.

If we plugged the values in the calculator:

Circle: $3.54\sqrt{A}$ (smallest perimeter)

Pentagon: $3.81\sqrt{A}$

Square: $4.00\sqrt{A}$

Triangle: $4.56\sqrt{A}$ (largest perimeter)

Why this order? The more sides a polygon has, the closer it gets to a circle (the most efficient shape). So: Circle < Pentagon < Square < Triangle.