Question: The area of an equilateral triangle is $ 25\sqrt{3} $ cm². If each side length is increased by 4 cm, by how many square centimeters does the area increase?
["Question: The area of an equilateral triangle is $ 25\sqrt{3} $ cm². If each side length is increased by 4 cm, by how many square centimeters does the area increase?", "---", "### Understanding Equilateral Triangle Area", "When exploring geometry, one commonly encountered shape is the equilateral triangle—a triangle with all three sides equal and all three angles measuring 60°. Calculating its area is straightforward using the formula:", "[\n\ ext{Area} = \frac{\sqrt{3}}{4} s^2\n]", "where ( s ) is the length of a side.", "Given that the area is $ 25\sqrt{3} $ cm², we can use this formula to find the original side length.", "---", "### Step 1: Solve for Original Side Length", "Set up the equation:", "[\n\frac{\sqrt{3}}{4} s^2 = 25\sqrt{3}\n]", "Divide both sides by $ \sqrt{3} $:", "[\n\frac{1}{4} s^2 = 25\n]", "Multiply both sides by 4:", "[\ns^2 = 100\n]", "Take the square root:", "[\ns = 10 \ ext{ cm}\n]", "So, the original side length is 10 cm.", "---", "### Step 2: Calculate New Side Length", "Each side is increased by 4 cm:", "[\ns_{\ ext{new}} = 10 + 4 = 14 \ ext{ cm}\n]", "---", "### Step 3: Calculate New Area", "Apply the area formula again:", "[\n\ ext{Area}_{\ ext{new}} = \frac{\sqrt{3}}{4} (14)^2 = \frac{\sqrt{3}}{4} \ imes 196 = 49\sqrt{3} \ ext{ cm}^2\n]", "---", "### Step 4: Find the Increase in Area", "Subtract the original area from the new area:", "[\n\Delta A = 49\sqrt{3} - 25\sqrt{3} = 24\sqrt{3} \ ext{ cm}^2\n]", "---", "### Conclusion", "When each side of an equilateral triangle with an area of $ 25\sqrt{3} $ cm² is increased by 4 cm, the area increases by:", "[\n\boxed{24\sqrt{3} \ ext{ cm}^2}\n]", "---", "Keywords: equilateral triangle area formula, side length increase area change, increase equilateral triangle area, how much does area increase triangle side added 4 cm, geometry problem solution, area calculation equilateral triangle", "Meta Description: Learn how to calculate the area of an equilateral triangle and determine the increase when each side is extended by 4 cm. Find step-by-step solution with final area increase of $ 24\sqrt{3} $ cm²."]
