Angles, Polygons and Geometrical Proof: Age 14-16

This is part of our collection of favourite rich tasks arranged by topic.

If you are a teacher, you can find the whole collection on our Secondary Curriculum teacher page.
Alternatively, if you are a student, you'll find the same problems on our Secondary Curriculum student page.


  • Tourism
    problem
    Favourite

    Tourism

    Age
    11 to 16
    Challenge level

    If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

  • Same length
    problem
    Favourite

    Same Length

    Age
    11 to 16
    Challenge level

    Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

  • Making sixty
    problem
    Favourite

    Making Sixty

    Age
    14 to 16
    Challenge level

    Why does this fold create an angle of sixty degrees?

  • circles in quadrilaterals
    problem
    Favourite

    Circles in Quadrilaterals

    Age
    14 to 16
    Challenge level

    Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

  • Isosceles Seven
    problem
    Favourite

    Isosceles Seven

    Age
    14 to 16
    Challenge level

    Is it possible to find the angles in this rather special isosceles triangle?

  • Small pepper seedlings in turquoise pots.
    problem
    Favourite

    Triangle Midpoints

    Age
    14 to 16
    Challenge level

    You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

  • Two Ladders
    problem
    Favourite

    Two Ladders

    Age
    14 to 16
    Challenge level

    Two ladders are propped up against facing walls. At what height do the ladders cross?

  • Sitting Pretty
    problem
    Favourite

    Sitting Pretty

    Age
    14 to 16
    Challenge level

    A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

  • Triangle in a Triangle
    problem
    Favourite

    Triangle in a Triangle

    Age
    14 to 16
    Challenge level

    Can you work out the fraction of the original triangle that is covered by the inner triangle?

  • Trapezium Four
    problem
    Favourite

    Trapezium Four

    Age
    14 to 16
    Challenge level

    The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

  • Nicely Similar
    problem
    Favourite

    Nicely Similar

    Age
    14 to 16
    Challenge level

    If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

  • The square under the hypotenuse
    problem
    Favourite

    The Square Under the Hypotenuse

    Age
    14 to 16
    Challenge level

    Can you work out the side length of a square that just touches the hypotenuse of a right angled triangle?

  • Napkin
    problem
    Favourite

    Napkin

    Age
    14 to 16
    Challenge level

    A napkin is folded so that a corner coincides with the midpoint of an opposite edge. Investigate the three triangles formed.

  • Angle Trisection
    problem
    Favourite

    Angle Trisection

    Age
    14 to 16
    Challenge level

    It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

  • Squirty
    problem
    Favourite

    Squirty

    Age
    14 to 16
    Challenge level

    Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

  • Partly Circles
    problem
    Favourite

    Partly Circles

    Age
    14 to 16
    Challenge level

    What is the same and what is different about these circle questions? What connections can you make?

  • Trapezium made of wooden tangram pieces, including a square and a parallelogram.
    problem
    Favourite

    Quad in Quad

    Age
    14 to 18
    Challenge level

    Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

  • Kite in a Square
    problem
    Favourite

    Kite in a Square

    Age
    14 to 18
    Challenge level

    Can you make sense of the three methods to work out what fraction of the total area is shaded?

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