Concepts in Action: Ratio between circumscribing and inscribed figures

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  • From triangular box: grey equilateral, four red equilateral triangles
  • From large hexagonal box: yellow equilateral, three yellow obtuse-angled
  • From small hexagonal box: six grey equilateral triangles
  • Triangle fraction insets 4/4, square fraction insets 4/4 (diagonal)



  1. Superimpose the four small red equilateral triangles on the grey equilateral triangle.
  2. Remove the three triangles at the vertices, leaving two equilateral triangles.
  3. An equilateral triangle inscribed in another equilateral triangle is 1/4 of it.
  4. The ratio is 1:4.
  5. Demonstrate the same experience using the metal insets.
  6. We can construct a chart to examine all regular polygons.

Number of sides Ratio

3 1/4
4 2/4
5 ? (between 2/4 and 3/4
6 3/4
etc... etc...
circle 4/4

   7.   Use the metal inset of the square.

   8.   Recall the value of the triangular pieces.

   9.   Put two pieces aside.

  10.  With the remaining two form an inscribed square.

  11.  The inscribed square is 2/4 of the circumscribing square.

  12.  We don't have any material to examine the pentagon.

  13.  For the hexagon, form H1 and H2 and superimpose them to recall the ratio of 3:4.

  14.  We have no more materials to examine the others.

  15.  When will the ratio be 4/4? When there is no difference between the circumscribing and inscribing figures, that is when the figures are circles.

  16.  We can conclude that the ratio for the pentagon will be somewhere between 2/4 and 3/4, and for the septagon and all the others, the ratio will be between 3/4 and 4/4.

Control Of Error

Points Of Interest