Draw the largest possible square inside an equilateral triangle, with one side of the square aligned with one side of the triangle. If the equilateral triangle has side length 6, find the side length of the square.

Rectangular area

side a = 3 units

side b = 2.60 units

Step-by-step explanation: See annex

We have an equilateral triangle (3 equal sides, three equal inside angles)

side length = 6

∠ ABC = ∠BCA = ∠CAB = 60⁰

tan 60⁰ = sin 60⁰/cos 60⁰ = √3/2 / 1/2

tan 60⁰ = √3

From annex

tan ∠ ABC = √3 = b/x then b = √3 * x

and a = 6 - 2*x (by symmetry)

Then

A = a*b

A (x) = (6 - 2*x) * √3 * x

A (x) = 6*√3 * x - 2*√3 * x²

Taking derivatives on both sides of the equation we get:

A' (x) = 6*√3 - 4*√3 * x ⇒ A' (x) = 0 ⇒ 6*√3 - 4*√3 * x = 0

x = 6/4 ⇒ x = 1.5

then a = 6 - 2*x ⇒ a = 6 - 3 ⇒ a = 3 (units)

and b = √3*x = 1.5 * √3 = 2.60 units

We get a rectangle (almost a square)