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At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Six Discs

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

Equilateral Areas

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

Towering Trapeziums

Age 14 to 16
Challenge Level

$OGH$ is an isosceles right-angled triangle:
 
Lines $AB$, $CD$, $EF$, and $GH$ are parallel.

Suppose the area of the smallest triangle $OAB$ is one square unit.
  • If lines $OC$ and $AB$ have the same length, calculate the area of trapezium $ABDC$.
  • If lines $OE$ and $CD$ also have the same length, calculate the area of trapezium $CDFE$.
  • If lines $OG$ and $EF$ also have the same length, calculate the area of trapezium $EFHG$.
Suppose that the chain of trapezia continued. 

What would be the area of the $n^{th}$ trapezium in the chain?

Can you explain your results?
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.
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