Question: An equilateral triangle has an area of $ 36\sqrt{3} $ cm². If each side is decreased by 4 cm, by how many square centimeters does the area decrease?
["Title: How the Area of an Equilateral Triangle Changes When Sidelength Decreases: A Step-by-Step Calculation", "Meta Description:\nDiscover how reducing each side of an equilateral triangle by 4 cm affects its area. Learn the precise difference in square centimeters when the triangle’s original area is $36\sqrt{3}~\ ext{cm}^2$, with a clear breakdown of calculations.", "---", "### Introduction\nGeometry problems often involve understanding how changing dimensions affect area—especially in regular shapes like equilateral triangles. In this article, we explore a practical question: If an equilateral triangle has an area of $36\sqrt{3}$ cm², and each side is decreased by 4 cm, by how many square centimeters does the area decrease? We’ll walk through the full derivation, ideal for students, teachers, or anyone interested in applied geometry.", "---", "### Step 1: Use the Area Formula for an Equilateral Triangle\nAn equilateral triangle of side length $s$ has area given by the formula:\n[\nA = \frac{\sqrt{3}}{4} s^2\n]\nWe are told the area is $36\sqrt{3}$ cm², so:\n[\n\frac{\sqrt{3}}{4} s^2 = 36\sqrt{3}\n]", "---", "### Step 2: Solve for the Original Side Length $s$\nDivide both sides by $\sqrt{3}$:\n[\n\frac{1}{4} s^2 = 36\n]\nMultiply both sides by 4:\n[\ns^2 = 144\n]\nTake the square root:\n[\ns = 12~\ ext{cm}\n]\nSo the original side length is 12 cm.", "---", "### Step 3: Calculate the New Side Length\nEach side is decreased by 4 cm:\n[\ns_{\ ext{new}} = 12 - 4 = 8~\ ext{cm}\n]", "---", "### Step 4: Compute the Original and New Areas\n- Original area (already given):\n[\nA_{\ ext{original}} = 36\sqrt{3}~\ ext{cm}^2\n]\n- New area with side 8 cm:\n[\nA_{\ ext{new}} = \frac{\sqrt{3}}{4} (8)^2 = \frac{\sqrt{3}}{4} \ imes 64 = 16\sqrt{3}~\ ext{cm}^2\n]", "---", "### Step 5: Find the Area Decrease\nSubtract the new area from the original:\n[\n\Delta A = A_{\ ext{original}} - A_{\ ext{new}} = 36\sqrt{3} - 16\sqrt{3} = 20\sqrt{3}~\ ext{cm}^2\n]", "---", "### Final Answer\nWhen each side of an equilateral triangle with an area of $36\sqrt{3}~\ ext{cm}^2$ is decreased by 4 cm, the area decreases by:\n[\n\boxed{20\sqrt{3}}~\ ext{cm}^2\n]", "---", "### Why This Matters\nUnderstanding how side length affects area is crucial in design, construction, and real-world measurements. This problem reinforces key geometric principles while providing a clear method for similar calculations in daily life and academic settings.", "---", "Keywords: equilateral triangle area, area decrease, equilateral triangle side change, geometry problems, $36\sqrt{3}$ triangle, side length reduction, square centimeter difference", "---", "Elevate your geometry skills with practical calculations—because every reduction matters! 📐✨"]
