# Isoperimetric Property of Equilateral Triangles

According to the Isoperimetric Theorem, circle has the maximum among all shapes with a given perimeter. Circle is probably the most *regular* of all plane shapes. Regularity also plays an important role in restricted families of plane figures. For example, it is known that

Among all triangles of given perimeter, the equilateral one has the largest area.

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Copyright © 1996-2018 Alexander Bogomolny

Among all triangles of given perimeter, the equilateral one has the largest area.

### Proof

The proof is based on Heron's formula

### Lemma

S² = s(s - a)(s - b)(s - c),

where a, b, c are the sides of a triangle, S its area, and s = (a + b + c)/2, the *semiperimeter*.

Since s is constant, the question is to maximize

(u + v + w) / 3 ≥ (uvw)^{1/3},

with equality only if u = v = w. For the three terms (s - a), (s - b),

(s / 3)³ ≥ (s - a)(s - b)(s - c) = S² / s,

so that S^{2} ≤ s^{4} / 27, with equality only when

To sum up, the area of a triangle with perimeter 2s never exceeds, s²/3^{3/2}; as is easy to verify, it exactly equals s²/3^{3/2} for an equilateral triangle with side 2s/3,s/3,1/3,2s/3,3s/2.

It must be understood (see the discussion of the general Isoperimetric Theorem) that our statement admits an equivalent formulation:

Among all triangles with given area, the equilateral one has the least perimeter.

(An alternative proof can be found elsewhere.)

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Copyright © 1996-2018 Alexander Bogomolny