The area of the original equilateral triangle is \( A_1 = \frac{\sqrt{3}}{4} s^2 \). The area of the smaller equilateral triangle with side \( \frac{s}{2} \) is \( A_2 = \frac{\sqrt{3}}{4} \left(\frac{s}{2}\right)^2 = \frac{\sqrt{3}}{4} \cdot \frac{s^2}{4} = \frac{\sqrt{3}}{16} s^2 \). The decrease in area is \( \Delta A = A_1 - A_2 = \frac{\sqrt{3}}{4} s^2 - \frac{\sqrt{3}}{16} s^2 = \left(\frac{4}{16} - \frac{1}{16}\right)\sqrt{3} s^2 = \frac{3\sqrt{3}}{16} s^2 \). The percentage decrease is:

["Understanding the Area Reduction: Percentage Decrease from an Original Equilateral Triangle", "Equilateral triangles are renowned for their symmetry and mathematical elegance. One of the most insightful properties of these triangles is how scaling their side length affects their area — particularly when the triangle is divided into a smaller equilateral triangle within the original.", "Consider an equilateral triangle with side length ( s ). Its area is given by:\n[\nA_1 = \frac{\sqrt{3}}{4} s^2\n]", "Now, imagine a smaller equilateral triangle formed by reducing the side length to half, ( \frac{s}{2} ). Its area becomes:\n[\nA_2 = \frac{\sqrt{3}}{4} \left(\frac{s}{2}\right)^2 = \frac{\sqrt{3}}{4} \cdot \frac{s^2}{4} = \frac{\sqrt{3}}{16} s^2\n]", "The decrease in area is calculated as:\n[\n\Delta A = A_1 - A_2 = \frac{\sqrt{3}}{4} s^2 - \frac{\sqrt{3}}{16} s^2\n]", "To compute this difference, convert terms to a common denominator:\n[\n\frac{\sqrt{3}}{4} = \frac{4\sqrt{3}}{16}, \quad \ ext{so} \quad \Delta A = \left(\frac{4\sqrt{3}}{16} - \frac{\sqrt{3}}{16}\right)s^2 = \frac{3\sqrt{3}}{16} s^2\n]", "To understand the reduction more precisely, calculate the percentage decrease relative to the original area:\n[\n\ ext{Percentage Decrease} = \left( \frac{\Delta A}{A_1} \right) \ imes 100% = \left( \frac{\frac{3\sqrt{3}}{16} s^2}{\frac{\sqrt{3}}{4} s^2} \right) \ imes 100%\n]", "Simplify the fraction:\n[\n\frac{\frac{3\sqrt{3}}{16}}{\frac{\sqrt{3}}{4}} = \frac{3\sqrt{3}}{16} \ imes \frac{4}{\sqrt{3}} = \frac{3 \cdot 4}{16} = \frac{12}{16} = \frac{3}{4}\n]", "Therefore:\n[\n\ ext{Percentage Decrease} = \frac{3}{4} \ imes 100% = 75%\n]", "This result reveals a profound insight: when reducing the side length of an equilateral triangle to half, the area decreases by 75%, not 50% as one might intuitively expect based on linear scaling. This quartile of the original area lost stems directly from the quadratic relationship in the area formula ( A \propto s^2 ). Understanding such geometric transformations deepens mathematical intuition and highlight the force of nonlinear scaling effects in uniform shapes.", "Key Takeaways:\n- Area of equilateral triangle: ( A = \frac{\sqrt{3}}{4} s^2 )\n- Halving the side length reduces area by 75%\n- This illustrates the power of squaring in area calculations", "Whether you're solving geometry problems or analyzing design proportions, recognizing how side length changes affect area is essential — and in the case of equilateral triangles, the impact is striking: a 75% area reduction with just a 50% length reduction."]