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.

Proof

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 S²/s = (s - a)(s - b)(s - c) for a + b + c = 2s. To this end we shall employ the arithmetic mean - geometric mean inequality which, for three terms u, v, w asserts that

(u + v + w) / 3 ≥ (uvw)1/3,

with equality only if u = v = w. For the three terms (s - a), (s - b), (s - c), that add up to s, we have

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

so that S2 ≤ s4 / 27, with equality only when s - a = s - b = s - c, i.e., for a = b = c.

To sum up, the area of a triangle with perimeter 2s never exceeds, s²/33/2; as is easy to verify, it exactly equals s²/33/2 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.)  • Equilateral Triangles on Sides of a Quadrilateral
• Euler Line Cuts Off Equilateral Triangle
• Four Incircles in Equilateral Triangle
• Problem in Equilateral Triangle
• Problem in Equilateral Triangle II
• Sum of Squares in Equilateral Triangle
• Triangle Classification
• Maximum Area Property of Equilateral Triangles
• Angle Trisectors on Circumcircle
• Equilateral Triangles On Sides of a Parallelogram
• Pompeiu's Theorem
• Circle of Apollonius in Equilateral Triangle
• 