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.
Polygon pictures
Can you work out how these polygon pictures were drawn, and use that to figure out their angles?
An equilateral triangular problem
Take an equilateral triangle and cut it into smaller pieces. What can you do with them?
Parallelogram it
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a parallelogram.
Quadrilaterals game
Tilted squares
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Square it
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
Triangles in circles
Can you find triangles on a 9-point circle? Can you work out their angles?
Semi-regular tessellations
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
Cyclic quadrilaterals
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
Hidden squares
Can you find the squares hidden on these coordinate grids?
Guess my quad
How many questions do you need to identify my quadrilateral?
Polygon rings
Join pentagons together edge to edge. Will they form a ring?
Completing quadrilaterals
We started drawing some quadrilaterals - can you complete them?
Star polygons
Draw some stars and measure the angles at their points. Can you find and prove a result about their sum?
Rhombus it
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a rhombus.
Square coordinates
A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?
Subtended angles
What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?
Right angles
Can you make a right-angled triangle on this peg-board by joining up three points round the edge?
Shapely pairs
A game in which players take it in turns to turn up two cards. If they can draw a triangle which satisfies both properties they win the pair of cards. And a few challenging questions to follow...
Property chart
A game in which players take it in turns to try to draw quadrilaterals (or triangles) with particular properties. Is it possible to fill the game grid?
Pythagoras proofs
Can you make sense of these three proofs of Pythagoras' Theorem?
Opposite vertices
Can you recreate squares and rhombuses if you are only given a side or a diagonal?
Quadrilaterals in a square
What's special about the area of quadrilaterals drawn in a square?
Which solids can we make?
Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?
Of all the areas
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
You may also be interested in this collection of activities from the STEM Learning website, that complement the NRICH activities above.