Sine, cosine, tangent

  • Circle Scaling
    problem

    Circle Scaling

    Age
    14 to 16
    Challenge level
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    Describe how to construct three circles which have areas in the ratio 1:2:3.
  • Pythagoras on a Sphere
    problem

    Pythagoras on a Sphere

    Age
    16 to 18
    Challenge level
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    Prove Pythagoras' Theorem for right-angled spherical triangles.
  • Lying and Cheating
    problem

    Lying and Cheating

    Age
    11 to 14
    Challenge level
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    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!

  • Dodecawhat
    problem

    Dodecawhat

    Age
    14 to 16
    Challenge level
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    Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.

  • A Scale for the Solar System
    problem

    A Scale for the Solar System

    Age
    14 to 16
    Challenge level
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    The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?

  • Round and Round
    problem

    Round and Round

    Age
    14 to 16
    Challenge level
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    Prove that the shaded area of the semicircle is equal to the area of the inner circle.

  • Circle Box
    problem

    Circle Box

    Age
    14 to 16
    Challenge level
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    It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

  • Eight Ratios
    problem

    Eight Ratios

    Age
    14 to 16
    Challenge level
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    Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found.

  • Moving Squares
    problem

    Moving Squares

    Age
    14 to 16
    Challenge level
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    How can you represent the curvature of a cylinder on a flat piece of paper?
  • Six Discs
    problem

    Six Discs

    Age
    14 to 16
    Challenge level
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    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?