Cylinder, Cone and Sphere (Surface Area and Volume)

351. The Toy Maker's Formula · Adding up curved areas and volumes

The toy's dimensions and formulas remain consistent as you drag the handles.

AOconehemispherethe dashed circle is the shared joining face — left out of the surface arear = 3r = 3h = 4h = 4l = 5l = 5Volume = ⅓πr²h + ⅔πr³ = 94.2Volume = ⅓πr²h + ⅔πr³ = 94.2Surface area = πr·l + 2πr² (no shared base) = 103.7Surface area = πr·l + 2πr² (no shared base) = 103.7rh
Many objects are combinations of solids (such as a cone on a hemisphere). Their total volume is the sum of the parts, while the surface area counts only the exposed faces — the shared joining faces are left out.

Stuck? Ask Guru

Selina ICSE: Cylinder, Cone and Sphere (Surface Area and Volume)

What this lesson covers

Try to break it

Drag R to set the shared radius (cone base = hemisphere) and H to set the cone's height. Exposed curved surface = πrℓ (cone) + 2πr² (hemisphere); volume = (1/3)πr²h + (2/3)πr³. The flat circle where the cone meets the hemisphere is hidden inside — it doesn't count toward the surface area.

How you build it

Build a cube with a cone on top: two square faces joined by four edges make the cube, then an ellipse base and two slant sides make the cone standing on it.

  • Square tool: draw the FRONT face of the cube.
  • Square tool: draw the BACK face — the same size, shifted up and to the right so the cube looks 3-D.
  • Segment tool: join the top-left corner of the front square to the top-left corner of the back square.
  • Segment tool: join the two squares' top-right corners.
  • Segment tool: join the two squares' bottom-left corners.
  • Segment tool: join the two squares' bottom-right corners. The four edges complete the cube.
  • Point tool: mark C at the centre of the cube's TOP face — the cone will sit here.
  • Circle tool: draw a circle on top of the cube, centred on C — the circular base of the cone.
  • Segment tool: from the LEFT edge of the ellipse, draw a line up to a point above C — the apex of the cone.
  • Segment tool: from the RIGHT edge of the ellipse, draw a line up to the SAME apex. The cone now stands on the cube — a combination of two solids.

The proof, step by step

Prove that the volume of a combined solid is the sum of the volumes of its parts.

  • The toy is formed by joining a cone and a hemisphere at their bases.
  • The common base is internal, so its area is excluded from the total surface area.
  • Total Surface Area = CSA of cone + CSA of hemisphere = πrl + 2πr²
  • Total Volume = Volume of cone + Volume of hemisphere = (1/3)πr²h + (2/3)πr³

Worked example

A toy is in the form of a cone mounted on a hemisphere with the same radius. The diameter of the base is 12 cm and its height is 8 cm. Determine the volume of the toy. (π = 3.14)

Radius r = 6 cm, h = 8 cm. Volume = 1/3 πr²h + 2/3 πr³ = 1/3 × 3.14 × 36 × 8 + 2/3 × 3.14 × 216 = 301.44 + 452.16 = 753.6 cm³.

  • 753.6 cm³ — correct
  • 800 cm³
  • 678.4 cm³
  • 900 cm³
Hold to talk

Subscription Status