In the textbook's derivation of the sphere surface area, the sphere is shown to have the same curved surface as a cylinder whose:

1. The derivation approximates the sphere by a stack of frustums and matches each frustum's curved area to a cylinder. 2. The full approximating cylinder has base radius equal to the sphere's radius r. 3. Its height equals 2r — that is, the sphere's diameter. 4. The cylinder's curved surface area is 2π × r × 2r = 4πr², equal to the sphere's surface area. 5. So the matching cylinder has radius r and height 2r. _Source: SCERT Kerala Std X Mathematics Part-2, Chapter 8-9 (pp 200-219, 2019 ed.)._