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

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Quadarc

Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.

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Get Cross

A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?

Six Discs

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Six circular discs, all of radius 1 unit, are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box.

  • Without using any calculations, compare the areas of the bases of boxes A and B and say which is the larger.
  • Use estimation to list the packings in order of increasing base area. Which are the hardest to put in order?
  • Now calculate the area of the base of each box and compare the correct order with your estimate.