2D shapes and their properties

problem

Circle Packing

Age

14 to 16

Challenge level

2 out of 3

Equal circles can be arranged so that each circle touches four or six others. What percentage of the plane is covered by circles in each packing pattern? ...
problem

Rhombus in Rectangle

Age

14 to 16

Challenge level

2 out of 3

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
problem

Sticky Tape

Age

14 to 16

Challenge level

2 out of 3

Work out the radius of a roll of adhesive tape.
problem

Angle to Chord

Age

14 to 16

Challenge level

2 out of 3

Weekly Problem 23 - 2008
A triangle has been drawn inside this circle. Can you find the length of the chord it forms?
problem

Towering Trapeziums

Age

14 to 16

Challenge level

2 out of 3

Can you find the areas of the trapezia in this sequence?
problem

Tied Up

Age

14 to 16

Challenge level

2 out of 3

How much of the field can the animals graze?
problem

Polycircles

Age

14 to 16

Challenge level

3 out of 3

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?
problem

From All Corners

Age

14 to 16

Challenge level

3 out of 3

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.
problem

Crescents and Triangles

Age

14 to 16

Challenge level

3 out of 3

Can you find a relationship between the area of the crescents and the area of the triangle?
problem

Squaring the Circle and Circling the Square

Age

14 to 16

Challenge level

3 out of 3

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.