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The cylinder, the cone and the sphere

Measures and solids, grade 6 Lesson 6 of 16Learning

The cylinder, the cone and the sphere

The cylinder

A prism and a pyramid have only flat faces. A cylinder, a cone and a sphere have a curved surface too.

A cylinder is a solid like a drink can, a roll of kitchen paper or a round candle:

  • The bases of the cylinder – two congruent discs, one exactly above the other. In the drawing they are coloured.
  • The lateral surface – the curved surface between the two bases.
  • The lateral surface meets each base in a curved line – the circle around the base.
The cylinder’s net

Cut a can’s label in a straight line, from base to base, and spread it out on the table: the label is a rectangle. Together with the two bases, this is the cylinder’s net:

  • One side of the rectangle is the cylinder’s height.
  • The other side went around the base exactly once, so it is exactly as long as the circle around the base.
The cone

A cone is a solid like an ice-cream cone, a party hat or a traffic cone:

  • The base of the cone – one disc.
  • The apex of the cone – the point above the base. In the drawing it is marked.
  • The lateral surface – the curved surface that rises from the base and ends at the apex.
The cone’s net

Cut the cone’s lateral surface in a straight line, from the apex to the base, and spread it out: it is a part of a disc, like an open fan. Together with the base, this is the cone’s net:

  • The fan’s point is the cone’s apex.
  • The fan’s arc goes around the base exactly once, so it is exactly as long as the circle around the base.
The sphere

A sphere is a solid like a basketball, a marble or a globe:

  • All the points on the sphere are at the same distance from one point – the centre of the sphere.
  • The whole sphere is one curved surface: it has no base, no edges and no vertices.
  • A sphere has no net: you cannot spread an orange peel flat without tearing it.
Exercise 1

Compare the cylinder, the cone and the sphere.
For each solid, write how many bases it has, whether it has a lateral surface, and whether it has a point that is an apex.

Exercise 2

Noga wraps the lateral surface of a can in coloured paper, with no overlap.
The can is 12 cm high.
The circle around the can’s base is 21 cm long.

a
How long is the rectangle she cuts, in cm? (its long side)

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b
    What is the rectangle’s area, in cm²?

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      Cross-sections

      Cut a solid with one straight cut, like a knife going through a cucumber. Where it is cut, a new face appears – the cross-section. In the drawing the cross-sections are coloured:

      1. A cylinder cut parallel to its base: the cross-section is a disc, equal to the base – like cucumber slices.
      2. A pyramid cut parallel to its base: the cross-section is a polygon of the same shape as the base, but smaller. The closer the cut is to the apex, the smaller the cross-section.
      3. A sphere, cut anywhere: the cross-section is a disc. The largest one goes through the sphere’s centre.
      Exercise 3

      What shape is each cross-section?
      Is it equal to the base, or smaller?

      Exercise 4

      A cake is a cylinder.
      It is cut from top to bottom, straight through the middle.
      What shape is the cross-section? Explain.