What is the formula for the surface area of an enclosed cylinder in terms of radius r, diameter d, and height h?

What is the formula for the surface area of an enclosed cylinder in terms of radius r, diameter d, and height h?

• A. 2πr^2 + πdh
• B. 2πr^2 + 2πrh
• C. πd^2 + 2πrh
• D. 2πrh + πd^2

Answer: (A)

Think of the cylinder’s surface area as the sum of the areas of the two bases and the curved side. The two circular bases give 2πr^2. The curved side, when unrolled, is a rectangle with width equal to the base circumference 2πr and height h, giving 2πrh. Since diameter d equals 2r, the lateral part 2πrh can be written as πdh. Put together, the total surface area is 2πr^2 + πdh. This matches the given option. The other forms either miss a base area or use diameter in a way that doesn’t reflect the correct base area or lateral area.

Think of the cylinder’s surface area as the sum of the areas of the two bases and the curved side. The two circular bases give 2πr^2. The curved side, when unrolled, is a rectangle with width equal to the base circumference 2πr and height h, giving 2πrh. Since diameter d equals 2r, the lateral part 2πrh can be written as πdh. Put together, the total surface area is 2πr^2 + πdh. This matches the given option. The other forms either miss a base area or use diameter in a way that doesn’t reflect the correct base area or lateral area.