Quad Areas

A Mathematics Lesson Starter Of The Day

Calculate the areas of all the possible quadrilaterals that can be constructed by joining together dots on this grid:

Teacher: Click on the dots above to show joining lines. Assume that the vertical and horizontal distances between adjacent dots is one unit.

A printable sheet for student use is available here.

Topics: Starter | Area | Geometry | Investigations | Mensuration | Shape

  • CB, Kent
  • I can find 10 different quadrilaterals (if you're allowed to count Big Square and Small Square). Areas - two 1s, two 1.5s, four 2s, one 3, one 4. Can't find the 2.5 at all. Am I missing something really obvious?
  • 8K3, Kettlethorpe High School
  • We found 11 which included the area of 2.5. (top left to top right to middle right to bottom middle to top left).
  • Snev, Holy Trinity, Crawley
  • There are 14 different quadrilaterals and their total area is 31.5 cm2 assuming the dots are 1 cm apart. (2x1)(4x1.5)(5x2)(1x2.5)(1x3)(1x4). The 2.5 cm2 quadrilateral joins dots 1,2,6 and 7.
  • Louise And Laura., Carnoustie High School
  • The problem was very challenging and we all enjoyed it. It was more challenging than Friday's.
  • Claire, Sam And Mr A, Alcester High School
  • We found 16 quadrilaterals, (5x1)(5x2)(3x1.5)(1x2.5)(1x3)(1x4)
    Only took us 3 minutes too (-ish!).
  • Nairn Spink, 2AY2, High School Of Dundee
  • We really enjoyed it and I thought it really got us thinking, we are really pleased to do all the staters of the day
  • Yr 8 Tops, Rego School
  • There should be quite a lot. My class managed about 30 so far.
  • Yr9 QTY, West Island School, Hong Kong
  • We enjoyed this - it kept us occupied for over 15mins.
  • Mrbeer, Essex
  • Surely with a parallelogram that crosses over the diagonals, the area is root 5 = 2.23?
  • Charles, Chelsea
  • My class found about 167 wow.
  • A And F, Dps
  • Our team got 32 different ways.
  • Jelissa, Massachusetts
  • It would be great if each quadrilateral could keep its own individual color, for a visual.
  • Steve, Carlsbad, CA
  • There are:
    70 convex quadrilaterals (not including the triangles)
    24 simple non-convex quadrilaterals
    48 triangles with an extra vertex on the midpoint of one side
    96 degenerate quadrilaterals where one edge overlays half of an adjacent edge
    140 complex quadrilaterals where two opposite sides intersect each other in a fifth point that is not a vertex of the quadrilateral
    The total number of quadrilaterals is 378, which is 126 ways to choose four vertices from nine possible points times three ways to arrange any of four vertices into a ring.
    The possible areas for the convex quadrilaterals are 1, 1.5, 2, 2.5, 3 or 4.
    The possible areas for the nonconvex quadrilaterals are 1 or 1.5.
    The possible areas for the triangles are 1 or 2.
    The possible areas for the degenerate quadrilaterals 0.5 or 1 (because it is half of the area of the corresponding triangle).
    The area of a complex quadrilateral is defined as the difference between the areas of the two triangles that are created. The possible areas for the complex quadrilaterals are 0, 0.5 or 1.

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