Question:** A cylinder has a height equal to twice its radius. If the total surface area of the cylinder is \(54\pi\) square units, find the radius of the cylinder.
["# Finding the Radius of a Cylinder: A Problem Solved", "When working with geometric shapes like cylinders, understanding how surface area relates to radius is essential. In this article, we’ll explore how to determine the radius of a cylinder when you know its height is twice its radius and the total surface area is given. Specifically, we’ll solve the problem: A cylinder has a height equal to twice its radius. If the total surface area of the cylinder is (54\pi) square units, find the radius.", "## Understanding the Geometry of a Cylinder", "A cylinder consists of two circular bases and a curved lateral surface. The total surface area (TSA) includes:\n- The area of the two bases (2 × πr²)\n- The lateral surface area (circumference × height = 2πrh)", "So, the formula for total surface area is:\n[\nTSA = 2\pi r^2 + 2\pi r h\n]", "Given in the problem, the height (h = 2r). This substitution simplifies the formula significantly, making it easier to solve for the radius.", "## Applying the Given Information", "Substitute (h = 2r) into the surface area formula:\n[\nTSA = 2\pi r^2 + 2\pi r (2r)\n]\n[\nTSA = 2\pi r^2 + 4\pi r^2\n]\n[\nTSA = 6\pi r^2\n]", "We are told the total surface area is (54\pi), so:\n[\n6\pi r^2 = 54\pi\n]", "## Solving for the Radius", "Divide both sides of the equation by (\pi) to eliminate it:\n[\n6r^2 = 54\n]\n[\nr^2 = \frac{54}{6} = 9\n]\n[\nr = \sqrt{9} = 3\n]", "Since radius cannot be negative, we take the positive root:\n[\nr = 3 \ ext{ units}\n]", "## Why This Matters: Practical Applications", "Understanding how to find a cylinder’s radius from its surface area and height relationship helps in design, manufacturing, and real-world engineering tasks. Whether manufacturing cylindrical containers, pipes, or packaging, precise measurements ensure material efficiency and structural reliability.", "## Conclusion", "Given a cylinder where height equals twice the radius and total surface area is (54\pi), solving algebraically confirms that the radius is (3) units. Mastering these steps strengthens geometric reasoning and problem-solving skills vital in math, science, and industry.", "---", "Keywords: cylinder radius, total surface area of a cylinder, solve for radius, geometry problem, cylinder surface area formula, (6\pi r^2 = 54\pi), math solution, problem-solving, cylindrical volume and surface area.", "Meta Description:\nLearn how to find the radius of a cylinder with height twice its radius and total surface area (54\pi). Solve the problem step-by-step using geometry formulas and algebraic reasoning. Ideal for students and math enthusiasts."]
