Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio.

Draw any triangle PQR. Find points A, B and C, one on each side of the triangle, such that the area of triangle ABC is a given fraction of the area of triangle PQR.

Can you work out the fraction of the original triangle that is covered by the green triangle?

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. . . .

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

One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

Can you find and prove the relationship between the area of a trapezium and the area of a triangle constructed within it?

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

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.

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Can you make sense of the three methods to work out the area of the kite in the square?

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?

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?

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.

Four rods are hinged at their ends to form a quadrilateral. How can you maximise its area?

Can you prove this formula for finding the area of a quadrilateral from its diagonals?

Do you have enough information to work out the area of the shaded quadrilateral?

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Can you find the area of a parallelogram defined by two vectors?

A trapezium is divided into four triangles by its diagonals. Can you work out the area of the trapezium?

A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two. . . .

Three fences of different lengths form three sides of an enclosure. What arrangement maximises the area?

Can you draw the height-time chart as this complicated vessel fills with water?