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Centres of mass of standard bodies

Edexcel IAL Mathematics · WME03 Mechanics 3

M3 extends centres of mass from the particles and laminas of M2 to solids and shells. Many results come from symmetry, and a few standard ones are listed in the formulae booklet.

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Symmetry first

A uniform body's centre of mass lies on every axis or plane of symmetry. So a uniform cylinder, cone or hemisphere has its centre of mass on its axis, and you only need one distance.

Standard results

Learn these, and check which ones your formulae booklet lists. Distances are along the axis of symmetry and angles are in radians.

Uniform bodyPosition of centre of mass
Solid hemisphere, radius r3r/8 from the centre of the plane face
Hemispherical shell, radius rr/2 from the centre
Solid cone or pyramid, height hh/4 above the base (3h/4 from the vertex)
Conical shell, height hh/3 above the base
Circular arc, radius r, angle 2α at the centrer sin α/α from the centre
Sector of a circle, radius r, angle 2α at the centre2r sin α/(3α) from the centre

Using the results

Always say where a distance is measured from. A common slip is measuring the cone's h/4 from the vertex instead of the base. The next two lessons show how integration proves these results, which Pearson can ask you to do.

Worked example

Find the distance of the centre of mass of a uniform solid hemisphere of radius 6 cm from the centre of its plane face.

  1. By symmetry it is on the axis.
  2. Distance = 3r/8 = 18/8.

Answer: 2.25 cm

Key idea: Use symmetry to find the axis, then a standard distance, measured from the right point.

Check yourself

  1. Find the distance of the centre of mass of a uniform solid hemisphere of radius 4 cm from the centre of its plane face.
    Show answer

    1.5 cm. 3r/8 = 1.5 cm.

  2. Find the distance of the centre of mass of a uniform hemispherical shell of radius 4 cm from its centre.
    Show answer

    2 cm. r/2 = 2 cm.

  3. A uniform solid cone has height 12 cm. How far above its base is its centre of mass?
    Show answer

    3 cm. h/4 = 3 cm.

  4. A uniform conical shell (no base) has height 12 cm. How far above the open end is its centre of mass?
    Show answer

    4 cm. h/3 = 4 cm.

  5. A uniform sector of a circle has radius 6 cm and angle π/3 at the centre. Find the distance of its centre of mass from the centre, to 3 significant figures.
    Show answer

    3.82 cm. 2α = π/3, so α = π/6. 2r sin α/(3α) = 2(6)(0.5)/(π/2) = 3.82 cm.

  6. A uniform wire is bent into an arc of a circle of radius 6 cm and angle π/3 at the centre. Find the distance of its centre of mass from the centre, to 3 significant figures.
    Show answer

    5.73 cm. r sin α/α = 6(0.5)/(π/6) = 5.73 cm.

  7. A uniform solid cone has height h. How far is its centre of mass from the vertex?
    • h/4
    • h/3
    • 3h/4
    • h/2
    Show answer

    3h/4. It is h/4 above the base, so h - h/4 = 3h/4 from the vertex.

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Sources

Last updated 8 November 2026. Spotted a mistake? Tell us.