Area - triangles, quadrilaterals, compound shapes

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

The farmers want to redraw their field boundary but keep the area the same. Can you advise them?

What's special about the area of quadrilaterals drawn in a square?

What are the possible areas of triangles drawn in a square?

Can you make sense of the three methods to work out what fraction of the total area is shaded?

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

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

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?