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Hemisphere on Top of a Cylinder

Surface Areas and Volumes · Class X

A common composite: a cylindrical vessel capped by a hemisphere. Compute combined surface area and capacity.

3.5
type a value
0.510
10
type a value
0.520
Live values
  • Cylinder CSA219.9115
  • Hemisphere CSA76.969
  • Combined outer surface area335.365
  • Combined volume474.6423
xy
  • Total volume as cylinder height varies

Formulas in this lab

  • Cylinder CSA
    2πrh2\pi r h
  • Hemisphere CSA
    2πr22\pi r^2
  • Combined outer surface area
    2πrh+2πr2+πr22\pi r h + 2\pi r^2 + \pi r^2
  • Combined volume
    πr2h+23πr3\pi r^2 h + \tfrac{2}{3}\pi r^3
Tip: Volume rises linearly with h; the y-intercept (h=0) is the bare hemisphere volume.

Frequently asked questions

▶How do we handle a composite solid?

For a cylindrical vessel of radius r and height h capped by a hemisphere of the same radius, total surface area is $2\pi r h + 2\pi r^2 + \pi r^2 = 2\pi r h + 3\pi r^2$ if the base is closed, and total volume is $\pi r^2 h + \frac{2}{3}\pi r^3$.

▶How do I use this lab?

Slide the common radius r and cylinder height h. The lab adds up the surface and volume of the two parts. Try r = 3.5, h = 10 to see how the hemisphere contributes a clear chunk to the capacity.

▶Common mistake on this topic

Students count the cylinder's top circle in the surface area when the hemisphere actually sits there. The shared circle is internal and must not be added. Always sketch the solid and mark which faces are exposed.

▶Where do we see this composite?

Modern hot-water geysers, lota-style water containers, and some temple gateposts are cylinders capped with a hemisphere. The same formula tells the manufacturer how much paint or steel to budget.