Sine, cosine, tangent

problem

Lying and Cheating

Age

11 to 14

Challenge level

2 out of 3

Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it!
problem

Dodecawhat

Age

14 to 16

Challenge level

1 out of 3

Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.
problem

A Scale for the Solar System

Age

14 to 16

Challenge level

1 out of 3

The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?
problem

Round and Round

Age

14 to 16

Challenge level

2 out of 3

Prove that the shaded area of the semicircle is equal to the area of the inner circle.
problem

Circle Box

Age

14 to 16

Challenge level

2 out of 3

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
problem

Eight Ratios

Age

14 to 16

Challenge level

2 out of 3

Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found.
problem

Moving Squares

Age

14 to 16

Challenge level

2 out of 3

How can you represent the curvature of a cylinder on a flat piece of paper?
problem

Six Discs

Age

14 to 16

Challenge level

3 out of 3

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?
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

Squ-Areas

Age

14 to 16

Challenge level

3 out of 3

Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more triangular areas are enclosed. What is the area of this convex hexagon?