Solution: The area of an equilateral triangle is $\frac{\sqrt{3}}{4} s^2$. Solving for original side $s$: $\frac{\sqrt{3}}{4} s^2 = 25\sqrt{3} \Rightarrow s^2 = 100 \Rightarrow s = 10\,\text{cm}$. New side: $14\,\text{cm}$. New area: $\frac{\sqrt{3}}{4} \cdot 14^2 = 49\sqrt{3}$. Increase: $49\sqrt{3} - 25\sqrt{3} = 24\sqrt{3}$. Thus, $\boxed{24\sqrt{3}}$.
["Solving for the Area Increase of an Equilateral Triangle: A Step-by-Step Breakdown", "Understanding the geometry of equilateral triangles goes beyond knowing their area formula—it’s about solving real-world problems involving shape transformations. This article explains how to calculate the area increase when the side length changes, with a clear, step-by-step solution.", "---", "### The Area Formula: Foundation of the Problem", "The area ( A ) of an equilateral triangle with side length ( s ) is given by:\n[\nA = \frac{\sqrt{3}}{4} s^2\n]", "This formula is derived from the properties of equilateral triangles, where all sides and angles are equal, and the height forms a natural split of the base.", "---", "### Step 1: Solve for the Original Side Length ( s )", "We are given that after scaling, the side becomes ( s = 14,\ ext{cm} ). Plug this into the area formula to find the new area:\n[\nA_{\ ext{new}} = \frac{\sqrt{3}}{4} \cdot (14)^2 = \frac{\sqrt{3}}{4} \cdot 196 = 49\sqrt{3}\n]", "So, the area with side ( 14,\ ext{cm} ) is ( 49\sqrt{3} ) square centimeters.", "---", "### Step 2: Calculate the Original Area", "Using the original side length ( s = 10,\ ext{cm} ) (derived from ( \frac{\sqrt{3}}{4} s^2 = 25\sqrt{3} )):\n[\nA_{\ ext{original}} = \frac{\sqrt{3}}{4} \cdot (10)^2 = \frac{\sqrt{3}}{4} \cdot 100 = 25\sqrt{3}\n]", "---", "### Step 3: Determine the Area Increase", "Subtract the original area from the new area:\n[\n\Delta A = 49\sqrt{3} - 25\sqrt{3} = 24\sqrt{3}\n]", "Thus, the increase in area is ( \boxed{24\sqrt{3}} ) square centimeters.", "---", "### Why This Calculation Matters", "Calculating how changes in side length affect the area is essential in architecture, engineering, and design. It explains the real-world impact of scaling geometric elements—such as expanding a triangular panel or refining a structural element—helping professionals make precise measurements and cost estimates.", "---", "### Summary", "- Area of an equilateral triangle: ( A = \frac{\sqrt{3}}{4} s^2 )\n- For side ( s = 14,\ ext{cm} ), area = ( 49\sqrt{3} )\n- For original side ( s = 10,\ ext{cm} ), area = ( 25\sqrt{3} )\n- Increase in area = ( 24\sqrt{3} )", "Understanding these calculations empowers precise problem-solving in mathematics and applied disciplines.", "---", "Key takeaway: Knowing how to reverse-engineer side lengths and compute area differences unlocks deeper geometric insight and practical utility—proving that even elegant formulas tell powerful stories behind every triangle."]
