Let the radius of the cylinder be \(r\). The height \(h\) is given as \(2r\). The total surface area \(S\) of the cylinder is given by:
["Title: Total Surface Area of a Cylinder with Radius (r) and Height (2r) Explained", "When studying geometry, one fundamental concept is understanding the surface area of a cylinder — a shape with two circular bases and a curved lateral surface. This article explores the total surface area (S) of a cylinder where the radius is (r) and the height (h = 2r).", "---", "### Understanding the Cylinder’s Structure", "A right circular cylinder consists of:\n- Two identical circular bases\n- A curved lateral (side) surface wrapped around the two bases", "Since the height (h) is given as (2r), the top and bottom circles sit perfectly aligned with the curved surface, and the height reaches twice the radius.", "---", "### Formula for Total Surface Area of a Cylinder", "The total surface area (S) includes:\n1. The area of the two circular bases\n2. The lateral surface area wrapping around the cylinder", "The mathematical formula is:", "[\nS = 2\pi r^2 + 2\pi r h\n]", "where:\n- (2\pi r^2) = total area of the two circular bases\n- (2\pi r h) = lateral surface area, calculated as circumference ((2\pi r)) times height ((h))", "---", "### Substituting (h = 2r) into the Formula", "With height (h = 2r), substitute into the surface area expression:", "[\nS = 2\pi r^2 + 2\pi r (2r)\n]", "Simplify:", "[\nS = 2\pi r^2 + 4\pi r^2 = 6\pi r^2\n]", "---", "### Final Result", "Thus, the total surface area of a cylinder with radius (r) and height (2r) is:", "[\n\boxed{S = 6\pi r^2}\n]", "---", "### Why This Matters", "Knowing the total surface area helps in practical applications — from designing fuel tanks and storage containers to calculating painting or material requirements. By setting (h = 2r), engineers and mathematicians optimize space usage and cost — balancing volume and surface with elegant efficiency.", "---", "### Summary", "- Radius: (r)\n- Height: (h = 2r)\n- Total surface area:\n[\nS = 2\pi r^2 + 2\pi r h = 6\pi r^2\n]", "Understanding how geometry principles like this apply to real-world cylindrical forms empowers precise planning and design in manufacturing, architecture, and education.", "---", "Keywords: cylinder surface area, total surface area of cylinder, radius (r), height (2r), formula (S = 6\pi r^2), geometry calculation, cylinder dimensions, math formula explanation."]
