An equilateral triangle has an area of \( 36\sqrt{3} \) square centimeters. If each side of the triangle is decreased by 2 cm, by how many square centimeters does the area decrease?
["Title: How Decreasing Each Side of an Equilateral Triangle Affects Its Area – A Step-by-Step Calculation", "An equilateral triangle with an area of ( 36\sqrt{3} ) cm² offers a classic geometry problem that combines area formulas, side-length relationships, and percentage-based area reduction. If each side of this triangle is decreased by 2 cm, we want to determine the exact decrease in area—critical information for applications in construction, design, and spatial planning.", "---", "### Understanding the Area of an Equilateral Triangle", "The formula for the area ( A ) of an equilateral triangle with side length ( s ) is:", "[\nA = \frac{\sqrt{3}}{4} s^2\n]", "We are given that the area equals ( 36\sqrt{3} ) cm². Setting up the equation:", "[\n\frac{\sqrt{3}}{4} s^2 = 36\sqrt{3}\n]", "To solve for ( s ), divide both sides by ( \sqrt{3} ):", "[\n\frac{1}{4} s^2 = 36\n]", "Multiply both sides by 4:", "[\ns^2 = 144\n]", "Taking the square root:", "[\ns = 12 \ ext{ cm}\n]", "So, the original triangle has each side measuring 12 cm.", "---", "### Calculating the New Area After Reducing Each Side", "Each side is decreased by 2 cm, so the new side length is:", "[\ns_{\ ext{new}} = 12 - 2 = 10 \ ext{ cm}\n]", "Now compute the area of the smaller equilateral triangle:", "[\nA_{\ ext{new}} = \frac{\sqrt{3}}{4} \ imes 10^2 = \frac{\sqrt{3}}{4} \ imes 100 = 25\sqrt{3} \ ext{ cm}^2\n]", "---", "### Determining the Area Decrease", "Subtract the new area from the original area:", "[\n\Delta A = 36\sqrt{3} - 25\sqrt{3} = 11\sqrt{3} \ ext{ cm}^2\n]", "---", "### Final Result: The Area Decreases by ( 11\sqrt{3} ) Square Centimeters", "This result demonstrates how reducing each side of an equilateral triangle by 2 cm reduces its area by ( 11\sqrt{3} ) cm²—useful for precise measurements in architectural design and mathematical education.", "---", "#### Summary\n- Original side: ( 12 ) cm\n- New side: ( 10 ) cm\n- Original area: ( 36\sqrt{3} ) cm²\n- New area: ( 25\sqrt{3} ) cm²\n- Area decrease: ( 11\sqrt{3} ) cm²", "Understanding this relationship helps in real-world applications such as scaling models, material estimation, and geometric problem solving.", "Keywords: equilateral triangle area, area decrease when side is reduced, geometric triangles, ( 36\sqrt{3} ) cm², side length change, area calculation formula, step-by-step geometry, square centimeters, triangle reduction problem", "---", "Stay tuned for more insightful geometry explanations and calculations."]
