This is part of our Secondary Curriculum collection of favourite rich tasks arranged by topic.
Scroll down to see the complete collection, or explore our subcollections on Perimeter and Area in two dimensions, and Surface Area and Volume in three dimensions.
problem
Favourite
Cyclic Quadrilaterals
Age
11 to 16
Challenge level
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
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?
problem
Favourite
Making sixty
Age
14 to 16
Challenge level
Why does this fold create an angle of sixty degrees?
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.
problem
Favourite
Isosceles Seven
Age
14 to 16
Challenge level
Is it possible to find the angles in this rather special isosceles triangle?
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?
problem
Favourite
Two Ladders
Age
14 to 16
Challenge level
Two ladders are propped up against facing walls. The end of the
first ladder is 10 metres above the foot of the first wall. The end
of the second ladder is 5 metres above the foot of the second wall.
At what height do the ladders cross?
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?
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?
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?
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?
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?
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?
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 .
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.
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.
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?
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You may also be interested in this collection of activities from the STEM Learning website, that complement the NRICH activities above.