Pythagoras' theorem

  • Folding in Half
    problem

    Folding in Half

    Age
    14 to 16
    Challenge level
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    How does the perimeter change when we fold this isosceles triangle in half?

  • Unusual Polygon
    problem

    Unusual Polygon

    Age
    14 to 16
    Challenge level
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    What is the perimeter of this unusually shaped polygon...

  • 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?
  • Two circles
    problem

    Two Circles

    Age
    14 to 16
    Challenge level
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    Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?

  • Ball Packing
    problem

    Ball Packing

    Age
    14 to 16
    Challenge level
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    If a ball is rolled into the corner of a room how far is its centre from the corner?

  • The Pillar of Chios
    problem

    The Pillar of Chios

    Age
    14 to 16
    Challenge level
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    Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

  • Crescents and triangles
    problem

    Crescents and Triangles

    Age
    14 to 16
    Challenge level
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    Can you find a relationship between the area of the crescents and the area of the triangle?

  • Squ-areas
    problem

    Squ-Areas

    Age
    14 to 16
    Challenge level
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    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?

  • Common Tangent
    problem

    Common Tangent

    Age
    14 to 16
    Challenge level
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    Two circles touch, what is the length of the line that is a tangent to both circles?

  • Interior Squares
    problem

    Interior Squares

    Age
    14 to 16
    Challenge level
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    Calculate the ratio of areas of these squares which are inscribed inside a semi-circle and a circle.