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Cylinder, cone, and sphere – surface area and volume

Mathematics3D GeometryAges 14–15

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Use with my class

A 3D model of a cylinder, cone, or sphere that can be rotated by dragging with the mouse or a finger. Drag the radius R and height h sliders to see the solid change along with the lateral surface area, total surface area, and volume formulas (Sxq = 2πRh, V = πR²h; Sxq = πRl, V = ⅓πR²h; S = 4πR², V = ⁴⁄₃πR³) shown with actual numbers. The 2D | 3D switch shows a true 3D cylinder, cone or sphere (WebGL) with the current R and h; drag to rotate, scroll or pinch to zoom.

Lesson: Cylinder, cone, and sphere: lateral surface area, total surface area, and volume

What it shows

Cylinders, cones and spheres are solids of revolution, and each area and volume formula can be linked to lengths visible on the model. A cylinder's curved surface unrolls into a rectangle measuring 2πR by h, giving an area of 2πRh, and its volume is πR²h. A cone's curved surface depends on the slant height l = √(R² + h²), not on h, and its volume is exactly one third of a cylinder with the same base and height. A sphere has area 4πR² and volume ⁴⁄₃πR³, which shows how area scales with R² and volume with R³.

How to use

Choose Cylinder, Cone or Sphere and change Base radius / radius R and Height h. Rotate the solid by dragging or with Auto-Rotate, and use the 2D | 3D button for a zoomable WebGL model. Keep R and h fixed while switching between Cylinder and Cone to compare the two volumes. Then double R and ask students to predict, before checking, how the volumes of the cylinder and the sphere change.

Parameters you can change

  • Solid Cylinder, Cone, Sphere
  • Base radius / radius R 1–10 cm
  • Height h (cylinder, cone) 1–20 cm
  • Auto-rotate

Questions to explore

  1. Why does the curved surface area of a cone use the slant height l instead of the height h?
  2. Compare a sphere of radius R with a cylinder of radius R and height 2R. Which has the larger volume?
  3. When R doubles, by what factor do the volume of the cylinder and the volume of the sphere change?

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