Sine, cosine, tangent

  • A Scale for the Solar System
    problem

    A Scale for the Solar System

    Age
    14 to 16
    Challenge level

    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

    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

    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

    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
    How can you represent the curvature of a cylinder on a flat piece of paper?
  • Cosines Rule
    problem

    Cosines Rule

    Age
    14 to 16
    Challenge level

    Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

  • Six Discs
    problem

    Six Discs

    Age
    14 to 16
    Challenge level
    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?
  • From all corners
    problem

    From All Corners

    Age
    14 to 16
    Challenge level

    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.

  • Squ-areas
    problem

    Squ-Areas

    Age
    14 to 16
    Challenge level

    Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more triangular areas are enclosed. What is the area of this convex hexagon?

  • Figure of Eight
    problem

    Figure of Eight

    Age
    14 to 16
    Challenge level

    On a nine-point pegboard a band is stretched over 4 pegs in a "figure of 8" arrangement. How many different "figure of 8" arrangements can be made ?

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