An equilateral triangle has a side length of \( s \) and its area is reduced by removing a smaller equilateral triangle from one corner, such that the side of the smaller triangle is \( \frac{s}{2} \). Determine the percentage decrease in the area of the triangle.

["Title: How Removing a Smaller Equilateral Triangle Reduces the Area of a Larger One – A Step-by-Step Analysis", "When working with equilateral triangles, precise area calculations reveal fascinating insights into geometric transformations. Imagine starting with a large equilateral triangle where each side measures ( s ). From one of its corners, a smaller equilateral triangle—with a side length of ( \frac{s}{2} )—is removed, leaving a truncated triangular shape. This process significantly alters the total area, and understanding the exact percentage decrease can enhance both mathematical intuition and practical design applications.", "In this article, we’ll explore the formula for the area of an equilateral triangle, compute both the original and reduced areas, and determine the percentage decrease in area caused by this geometric removal.", "---", "### Understanding the Area of an Equilateral Triangle", "An equilateral triangle with side length ( s ) has a well-known formula for its area:", "[\n\ ext{Area} = \frac{\sqrt{3}}{4} s^2\n]", "This formula derives from splitting the triangle into two right triangles using its altitude, then applying the Pythagorean theorem and basic trigonometry.", "---", "### Step 1: Area of the Original Triangle", "Using the formula, the area of the original triangle with side ( s ) is:", "[\nA_{\ ext{original}} = \frac{\sqrt{3}}{4} s^2\n]", "This represents 100% of the initial triangular area.", "---", "### Step 2: Area of the Smaller Removed Triangle", "A smaller equilateral triangle is removed from one corner, with side length ( \frac{s}{2} ). Applying the same area formula:", "[\nA_{\ ext{small}} = \frac{\sqrt{3}}{4} \left( \frac{s}{2} \right)^2 = \frac{\sqrt{3}}{4} \cdot \frac{s^2}{4} = \frac{\sqrt{3}}{16} s^2\n]", "This smaller triangle constitutes a fraction of the original area.", "---", "### Step 3: Area After Removal", "The remaining area is the difference:", "[\nA_{\ ext{remaining}} = A_{\ ext{original}} - A_{\ ext{small}} = \frac{\sqrt{3}}{4} s^2 - \frac{\sqrt{3}}{16} s^2\n]", "Factor out ( \frac{\sqrt{3}}{16} s^2 ):", "[\nA_{\ ext{remaining}} = \frac{\sqrt{3}}{16} s^2 \left(4 - 1\right) = \frac{\sqrt{3}}{16} s^2 \cdot 3 = \frac{3\sqrt{3}}{16} s^2\n]", "---", "### Step 4: Percentage Decrease in Area", "To find the percentage reduction, compute the ratio of the area lost to the original area:", "[\n\ ext{Percentage Decrease} = \left( \frac{A_{\ ext{original}} - A_{\ ext{remaining}}}{A_{\ ext{original}}} \right) \ imes 100%\n]", "Substitute the expressions:", "[\n= \left( \frac{ \frac{\sqrt{3}}{4} s^2 - \frac{3\sqrt{3}}{16} s^2 }{ \frac{\sqrt{3}}{4} s^2 } \right) \ imes 100%\n]", "Factor ( \frac{\sqrt{3}}{4} s^2 ) from numerator:", "[\n= \left( \frac{ \frac{1}{4} - \frac{3}{16} }{ \frac{1}{4} } \right) \ imes 100% = \left( \frac{ \frac{4}{16} - \frac{3}{16} }{ \frac{1}{4} } \right) \ imes 100% = \left( \frac{ \frac{1}{16} }{ \frac{1}{4} } \right) \ imes 100%\n]", "Simplify the fraction:", "[\n\frac{1/16}{1/4} = \frac{1}{16} \cdot \frac{4}{1} = \frac{4}{16} = \frac{1}{4}\n]", "Now multiply by 100%:", "[\n\ ext{Percentage Decrease} = \frac{1}{4} \ imes 100% = 25%\n]", "---", "### Conclusion", "By removing a smaller equilateral triangle from one corner—specifically, one with half the side length—a reduction of exactly 25% is observed in the area of the original equilateral triangle. This elegant geometric transformation underscores how proportional scaling affects areas and serves as a powerful example in architectural design, computing, and spatial reasoning.", "Understanding such proportionate changes helps artists, engineers, and educators visualize and manipulate shapes with precision and creativity.", "---", "Keywords: equilateral triangle area, area reduction, 25% decrease, geometric transformation, equilateral triangle formula, side length ( s ), percentage decrease in area, triangle removal process.", "Meta Description: Discover how removing a smaller equilateral triangle with half the side length from a larger triangle reduces its area by exactly 25%. Learn the step-by-step calculation with area formulas and percentage analysis."]