Area - triangles, quadrilaterals, compound shapes

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 trapeziums each of equal area. How could he do this?

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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.

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

Determine the total shaded area of the 'kissing triangles'.

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?

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

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