Pythagoras' theorem

  • In a Spin
    problem

    In a Spin

    Age
    14 to 16
    Challenge level
    What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?
  • Circumnavigation
    problem

    Circumnavigation

    Age
    14 to 16
    Challenge level
    The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.
  • Circle Scaling
    problem

    Circle Scaling

    Age
    14 to 16
    Challenge level
    Describe how to construct three circles which have areas in the ratio 1:2:3.
  • Spherical triangles on very big spheres
    problem

    Spherical Triangles on Very Big Spheres

    Age
    16 to 18
    Challenge level
    Shows that Pythagoras for Spherical Triangles reduces to Pythagoras's Theorem in the plane when the triangles are small relative to the radius of the sphere.
  • A black goat with horns and white facial markings.
    problem

    The Old Goats

    Age
    11 to 14
    Challenge level

    A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't fight each other but can reach every corner of the field?

  • Small pepper seedlings in turquoise pots.
    problem

    Rectangle Dissection

    Age
    11 to 14
    Challenge level

    Weekly Problem 2 - 2009
    The 16 by 9 rectangle is cut as shown. Rearrange the pieces to form a square. What is the perimeter of the square?

  • Square Pegs
    problem

    Square Pegs

    Age
    11 to 14
    Challenge level

    Which is a better fit, a square peg in a round hole or a round peg in a square hole?

  • Isosceles
    problem

    Isosceles

    Age
    11 to 14
    Challenge level

    Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

  • Trice
    problem

    Trice

    Age
    11 to 14
    Challenge level

    ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

  • A chordingly
    problem

    A Chordingly

    Age
    11 to 14
    Challenge level

    Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

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