List

Geometrical Reasoning

Creating convincing arguments or "proofs" to show that statements are always true is a key mathematical skill. The problems in this feature offer you the chance to explore geometrical properties, make conjectures and create proofs to show that these are always true.

Many of the problems in this feature include proof sorting activities which challenge you to rearrange statements in order to recreate clear, rigorous proofs.

Plus magazine has a selection of interesting articles exploring proofs in which pictures play an important role.

problem

Circumference angles

Age

11 to 16

Challenge level

Can you prove the angle properties described by some of the circle theorems?

problem

Cyclic quadrilaterals proof

Age

11 to 16

Challenge level

Can you prove that the opposite angles of cyclic quadrilaterals add to $180^\circ$?

problem

Favourite

Pythagoras proofs

Age

11 to 16

Challenge level

Can you make sense of these three proofs of Pythagoras' Theorem?

problem

Pentakite

Age

14 to 18

Challenge level

Given a regular pentagon, can you find the distance between two non-adjacent vertices?

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

Matter of scale

Age

14 to 16

Challenge level

Can you prove Pythagoras' Theorem using enlargements and scale factors?

problem

Overlap

Age

14 to 16

Challenge level

A red square and a blue square overlap. Is the area of the overlap always the same?

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

The converse of Pythagoras

Age

14 to 18

Challenge level

Can you prove that triangles are right-angled when $a^2+b^2=c^2$?

We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of these resources.