Question: An equilateral triangle’s side is reduced by 3 cm, decreasing its area by $ 15\sqrt{3} \, \text{cm}^2 $. What was the original side length?

["Asked Question & Solution: How to Find the Original Side Length of an Equilateral Triangle When Its Area Decreases by $15\sqrt{3} , \ ext{cm}^2$ After Reducing Each Side by 3 cm", "---", "Understanding Area Changes in Equilateral Triangles: A Clear Approach", "When the side length of an equilateral triangle is reduced by a fixed amount, its area decreases in a predictable mathematical way. This article explains how to solve a common geometry problem: An equilateral triangle’s side is reduced by 3 cm, decreasing its area by $15\sqrt{3} , \ ext{cm}^2$. What was the original side length?", "---", "### The Geometry of Equilateral Triangles", "The area $ A $ of an equilateral triangle with side length $ s $ cm is given by the formula:", "$$\nA = \frac{\sqrt{3}}{4} s^2\n$$", "This formula comes from dividing the triangle into 30-60-90 right triangles and using trigonometric relationships.", "---", "### Applying the Problem Conditions", "Let the original side length be $ s $ cm.", "- After reducing each side by 3 cm, the new side becomes $ s - 3 $ cm.\n- The original area is $ \frac{\sqrt{3}}{4} s^2 $\n- The new area is $ \frac{\sqrt{3}}{4} (s - 3)^2 $", "The decrease in area is given as $ 15\sqrt{3} , \ ext{cm}^2 $, so:", "$$\n\frac{\sqrt{3}}{4} s^2 - \frac{\sqrt{3}}{4} (s - 3)^2 = 15\sqrt{3}\n$$", "---", "### Step-by-Step Algebraic Solution", "Step 1: Factor out $ \frac{\sqrt{3}}{4} $:", "$$\n\frac{\sqrt{3}}{4} \left( s^2 - (s - 3)^2 \right) = 15\sqrt{3}\n$$", "Step 2: Cancel $ \sqrt{3} $ from both sides:", "$$\n\frac{1}{4} \left( s^2 - (s - 3)^2 \right) = 15\n$$", "Step 3: Expand $ (s - 3)^2 = s^2 - 6s + 9 $:", "$$\n\frac{1}{4} \left( s^2 - (s^2 - 6s + 9) \right) = 15\n$$", "$$\n\frac{1}{4} \left( s^2 - s^2 + 6s - 9 \right) = 15\n$$", "$$\n\frac{1}{4} (6s - 9) = 15\n$$", "Step 4: Multiply both sides by 4:", "$$\n6s - 9 = 60\n$$", "Step 5: Solve for $ s $:", "$$\n6s = 69 \quad \Rightarrow \quad s = \frac{69}{6} = 11.5\n$$", "---", "### Final Answer", "The original side length of the equilateral triangle is 11.5 cm, or $ \frac{23}{2} $ cm.", "---", "### Why This Problem Matters (SEO & User Intent)", "This type of question is frequently searched by students and math learners looking to master triangle area formulas and algebraic manipulation. Optimizing for search engines involves focusing on:", "- Clear, structured explanations\n- Easy-to-follow steps\n- Relevant keywords: equilateral triangle area change, side length solved, ring area decrease formula, inequilateral triangle area reduction", "By answering precisely and explaining how the solution is derived, this article helps users not only find the correct answer but also deepen their understanding of geometric principles tied to perimeter and area relationships.", "---", "Summary:\nReducing each side of an equilateral triangle by 3 cm decreases its area by $15\sqrt{3} , \ ext{cm}^2$. Using the area formula $ A = \frac{\sqrt{3}}{4} s^2 $, solving the equation reveals the original side length was 11.5 cm.", "---", "Keywords: equilateral triangle side length, area change formula, solve geometry problem, how to find original triangle side, triangle area reduction, algebraic geometry, $ 15\sqrt{3} $ geometry"]