Parallelogram 15 Level 1 14 Dec 2023Shapes of constant width

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteau word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.
  • Tackle each Parallelogram in one go. Don’t get distracted.
  • Finish by midnight on Sunday if your whole class is doing parallelograms.
  • Your score & answer sheet will appear immediately after you hit SUBMIT.
  • Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1. Does a wheel have to be circular?

Watch this video where Steve Mould shows an array of shapes with a peculiar property:

(If you have problems watching the video, right click to open it in a new window)

2 marks

1.1 Which of these shapes is a solid of constant width?

  • Hexagon
  • Circle
  • Triangle
  • Square
  • (Not answered)
2 marks

1.2 A triangle can be transformed into a Reuleaux triangle by rounding the edges, which set of polygons can this be done for?

  • All polygons
  • Only regular polygons
  • Polygons with an even number of edges
  • Polygons with an odd number of edges
  • (Not answered)
2 marks

1.3 Why are coins made with constant width?

  • So people aren’t hurt by the sharp edges in their pocket
  • It is cheaper
  • So they can roll in vending machines
  • (Not answered)

2. Cutting Shapes

2 marks

2.1 If a regular hexagon is cut in half, from point to point, it will produce two of which shape:

  • Pentagon
  • Triangle
  • Trapezium
  • Tetrahedron
  • (Not answered)
2 marks

2.2 A regular tetrahedron with edges of length 6 cm has each corner cut off to produce the solid shown.

The triangular faces are all equilateral triangles, but not necessarily the same size.

What is the total length of the edges of the resulting solid?

Correct Solution: 36 cm

Show Hint (–1 mark)
–1 mark

Do you notice any lengths which are the same?

Although the tetrahedron (triangle based pyramid) has been truncated, the total edge length remains the same.

6 edges at 6cm each = 36cm

Before you hit the SUBMIT button, here are some quick reminders:

  • You will receive your score immediately, and collect your reward points.
  • You might earn a new badge... if not, then maybe next week.
  • Make sure you go through the solution sheet – it is massively important.
  • A score of less than 50% is ok – it means you can learn lots from your mistakes.
  • The next Parallelogram is next week, at 3pm on Thursday.
  • Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelograms.

Cheerio, Simon & Ayliean.