Notes to teachers:
The usual approach to area formulas is to derive the triangle formula from the parallelogram formula. That requires reference to a substantial number of theorems from plane geometry. Furthermore, many sources give an incorrect proof that parallelogram area = bh, since they assume that the parallel sides being used as bases have a common perpendicular.
The tentative outline below avoids both difficulties. When completed, it will explain precisely what is meant by 'base' and 'height,' words whose definitions are not always presented carefully.
• In the previous screen, you learned that it's hard to figure out area by counting how many unit squares fit in a region. It's not hard to figure out simple formulas for the area of triangles and quadrilaterals. Click in order on the links below to see how.
4a. Area of rectangles
4b. Area of right triangles
4c. Area of triangles
4d. Area of triangles whose alititude falls inside the base
4e. Area of triangles whose alititude doesn't meet the base
4f. Area of parallelograms
4g. Area of trapezoids