Regular polygons and circles

  • Cube Paths
    problem

    Cube paths

    Age
    11 to 14
    Challenge level
    filled star filled star empty star
    Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B?
  • Circle Packing
    problem

    Circle packing

    Age
    14 to 16
    Challenge level
    filled star filled star empty star
    Equal circles can be arranged so that each circle touches four or six others. What percentage of the plane is covered by circles in each packing pattern? ...
  • Star Gazing
    problem

    Star gazing

    Age
    14 to 16
    Challenge level
    filled star empty star empty star
    Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.
  • From all corners
    problem

    From all corners

    Age
    14 to 16
    Challenge level
    filled star filled star filled star
    Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.
  • Square Pegs
    problem

    Square pegs

    Age
    11 to 14
    Challenge level
    filled star filled star filled star
    Which is a better fit, a square peg in a round hole or a round peg in a square hole?
  • Hexagon Transformations
    problem

    Hexagon transformations

    Age
    7 to 11
    Challenge level
    filled star filled star empty star
    Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!
  • Escriptions
    problem

    Escriptions

    Age
    16 to 18
    Challenge level
    filled star empty star empty star
    For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.
  • Retracircles
    problem

    Retracircles

    Age
    16 to 18
    Challenge level
    filled star empty star empty star
    Four circles all touch each other and a circumscribing circle. Find the ratios of the radii and prove that joining 3 centres gives a 3-4-5 triangle.
  • Get Cross
    problem

    Get cross

    Age
    14 to 16
    Challenge level
    filled star filled star empty star
    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?
  • Fitting In
    problem

    Fitting in

    Age
    14 to 16
    Challenge level
    filled star empty star empty star
    The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ