2D shapes and their properties

  • Pent
    problem

    Pent

    Age
    14 to 18
    Challenge level
    filled star filled star empty star
    The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.
  • Hex
    problem

    Hex

    Age
    11 to 14
    Challenge level
    filled star empty star empty star
    Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.
  • Tricircle
    problem

    Tricircle

    Age
    14 to 16
    Challenge level
    filled star filled star empty star
    The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and two sides of the triangle. If the small circle has radius 1 unit find the radius of the larger circle.
  • Rhombus in Rectangle
    problem

    Rhombus in rectangle

    Age
    14 to 16
    Challenge level
    filled star filled star empty star
    Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
  • 2001 Spatial Oddity
    problem

    2001 spatial oddity

    Age
    11 to 14
    Challenge level
    filled star filled star filled star
    With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
  • The medieval octagon
    problem

    The medieval octagon

    Age
    14 to 16
    Challenge level
    filled star filled star empty star
    Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
  • Holly
    problem

    Holly

    Age
    14 to 16
    Challenge level
    filled star filled star empty star
    The ten arcs forming the edges of the "holly leaf" are all arcs of circles of radius 1 cm. Find the length of the perimeter of the holly leaf and the area of its surface.
  • Lying and Cheating
    problem

    Lying and cheating

    Age
    11 to 14
    Challenge level
    filled star filled star empty star
    Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it!
  • Three four five
    problem

    Three four five

    Age
    14 to 16
    Challenge level
    filled star filled star empty star
    Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
  • Floored
    problem

    Floored

    Age
    14 to 16
    Challenge level
    filled star filled star empty star
    A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?