Solution: The lateral surface area of a cylinder is $ 2\pi r h $. The area of a circle with radius $ 2r $ is $ \pi (2r)^2 = 4\pi r^2 $. Setting them equal: $ 2\pi r h = 4\pi r^2 $. Cancel $ 2\pi r $ to get $ h = 2r $. Thus, the ratio $ \frac{h}{r} = \boxed{2} $.

February 22, 2026

$Solution: The lateral surface area of a cylinder is $ 2\pi r h $. The area of a circle with radius $ 2r $ is $ \pi (2r)^2 = 4\pi r^2 $. Setting them equal: $ 2\pi r h = 4\pi r^2 $. Cancel $ 2\pi r $ to get $ h = 2r $. Thus, the ratio $ \frac{h}{r} = \boxed{2} $.$

["Understanding the Lateral Surface Area of a Cylinder: A Key Geometric Insight", "When studying geometry, one essential concept is the lateral surface area of a cylinder. This measurement represents the area of the side surface—excluding the top and bottom bases—making it crucial in fields like engineering, architecture, and packaging design.", "### What Is the Lateral Surface Area of a Cylinder?", "The lateral surface area (( A_{\ ext{lateral}} )) of a cylinder is calculated using the formula:", "[\nA_{\ ext{lateral}} = 2\pi r h\n]", "where\n- ( r ) is the radius of the cylinder’s base,\n- ( h ) is the height or length of the cylinder.", "This formula arises from "unrolling" the curved surface into a rectangle: the circumference of the base (( 2\pi r )) times the height (( h )) gives the total area of the lateral face.", "### A Comparison with Area of a Circle", "Interestingly, if we compare this lateral area to the area of a full circle with radius ( 2r ), we get:", "[\n\ ext{Area of circle with radius } 2r = \pi (2r)^2 = 4\pi r^2\n]", "Now, set this equal to the cylinder’s lateral surface area:", "[\n2\pi r h = 4\pi r^2\n]", "Here, both expressions represent the same geometric quantity—so we can solve for ( h ) in terms of ( r ).", "### Solving for the Height-to-Radius Ratio", "Divide both sides by ( 2\pi r ) (assuming ( r <br/>\neq 0 )):", "[\nh = \frac{4\pi r^2}{2\pi r} = 2r\n]", "Thus, the height ( h ) is exactly twice the radius:", "[\nh = 2r\n]", "Dividing both sides by ( r ) gives the key ratio:", "[\n\frac{h}{r} = \boxed{2}\n]", "### Why This Matters", "This simple ratio is vital for practical applications—from designing containers and tanks to optimizing material usage. It confirms a proportional relationship between height and radius that ensures efficient surface area coverage with minimal waste.", "### Conclusion", "Recognizing that the lateral surface area of a cylinder and the full area of a scaled circle share a proportional relationship offers powerful insight. The derivation ( h = 2r ), leading to ( \frac{h}{r} = 2 ), reinforces fundamental geometric principles and enhances problem-solving skills in technical fields.", "[\n\boxed{2}\n]"]

Solution: The lateral surface area of a cylinder is $ 2\pi r h $. The area of a circle with radius $ 2r $ is $ \pi (2r)^2 = 4\pi r^2 $. Setting them equal: $ 2\pi r h = 4\pi r^2 $. Cancel $ 2\pi r $ to get $ h = 2r $. Thus, the ratio $ \frac{h}{r} = \boxed{2} $.

Related Articles

Trending Articles