2D shapes and their properties

problem
Pentagonal

Age

14 to 16

Challenge level

Can you prove that the sum of the distances of any point inside a square from its sides is always equal (half the perimeter)? Can you prove it to be true for a rectangle or a hexagon?
problem
Square coordinates

Age

11 to 14

Challenge level

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?
problem
Efficient cutting

Age

11 to 14

Challenge level

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.
problem
Making shapes

Age

5 to 7

Challenge level

Arrange any number of counters from these 18 on the grid to make a rectangle. What numbers of counters make rectangles? How many different rectangles can you make with each number of counters?
problem
Square it

Age

11 to 16

Challenge level

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.
problem
Salinon

Age

14 to 16

Challenge level

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
problem
Rolling around

Age

11 to 14

Challenge level

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
problem
Overlapping circles

Age

7 to 11

Challenge level

What shaped overlaps can you make with two circles which are the same size?
problem
Lunar angles

Age

16 to 18

Challenge level

What is the sum of the angles of a triangle whose sides are circular arcs on a flat surface? What if the triangle is on the surface of a sphere?
problem
Circles in circles

Age

16 to 18

Challenge level

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.