A square and a rectangle have the same perimeter. The rectangle's length is twice its width. If the square's side is 10 meters, what is the rectangle's area?
["A Square and a Rectangle with Equal Perimeters: Understanding Dimensions and Area", "When studying geometry, one common question arises: Can a square and a rectangle have the same perimeter? The answer is yes—especially when carefully selected dimensions align. In this article, we’ll explore a real-world example where a square with a side length of 10 meters shares the same perimeter as a rectangle whose length is twice its width. We’ll determine the rectangle’s area—showing how geometric relationships help optimize space efficiently.", "---", "### Understanding Square and Rectangle Perimeters", "A square has all four sides equal, and its perimeter is simply ( P = 4 \ imes \ ext{side} ). With a side length of 10 meters:", "[\n\ ext{Square perimeter} = 4 \ imes 10 = 40 \ ext{ meters}\n]", "A rectangle has two equal lengths and two equal widths. Its perimeter is:", "[\nP = 2 \ imes (\ ext{length} + \ ext{width})\n]", "Given the rectangle’s length is twice its width (( L = 2W )), its perimeter becomes:", "[\nP = 2 \ imes (2W + W) = 2 \ imes 3W = 6W\n]", "Since both shapes share the same perimeter (40 meters), we set:", "[\n6W = 40 \implies W = \frac{40}{6} = \frac{20}{3} \approx 6.67 \ ext{ meters}\n]", "Thus, the length is:", "[\nL = 2W = 2 \ imes \frac{20}{3} = \frac{40}{3} \approx 13.33 \ ext{ meters}\n]", "---", "### Calculating the Rectangle’s Area", "The area ( A ) of a rectangle is:", "[\nA = \ ext{length} \ imes \ ext{width} = L \ imes W = \frac{40}{3} \ imes \frac{20}{3} = \frac{800}{9} \approx 88.89 \ ext{ square meters}\n]", "This area perfectly matches the rectangle’s space, given its same perimeter as the square.", "---", "### Key Takeaway", "Even though a square and rectangle differ in shape, their perimeters can match through adjusted dimensions. With a side of 10 meters, a rectangle with length twice its width achieves 40 meters in perimeter—and a balanced area of ( \frac{800}{9} ) m². This exemplifies how geometry provides flexibility and efficiency in design and construction—balancing form, space, and measurement.", "---", "So, if a square has a 10-meter side, the rectangle with length twice its width and same 40-meter perimeter has an area of ( \frac{800}{9} ) square meters—proving math symmetry and real-world applicability."]
