A quadrilateral is a figure made up of four straight sides. The most familiar examples of quadrilaterals are squares and rectangles. In this assignment we will discover formulas for the area of three other types of quadrilaterals: trapezoids, parallelograms, and rhombuses. A trapezoid is a quadrilateral with a pair of parallel sides. A parallelogram has two pairs of parallel sides. Finally, a rhombus is a parallelogram with four sides of equal length. Note that a parallelogram is a special case of a trapezoid. Therefore, if we find a formula for the area of a trapezoid, then that formula will hold for parallelograms and rhombuses. In fact, squares and a rectangles are also special types of parallelograms, so the formula we will find will work for them also.
We will see that the area of a trapezoid is 1/2 × (x + y) × h, where x and y are the lengths of the two parallel sides and h is the corresponding height, as drawn in the picture above.
Problem 1. Draw a trapezoid, labelling the parallel sides and the height as in the picture above, and cut it out. Draw an identical copy of your trapezoid and cut it out also. Cut the trapezoids into pieces in such a way that you can rearrange the pieces to obtain two rectangles, one whose base is x and height is h and the other whose base is y and height is h. Using the formula for the area of a rectangle, determine the total area of the two rectangles, and then use it to determine the area of the original trapezoid. Draw the original trapezoid and the resulting rectangles, and explain why your construction shows that the area of a trapezoid is given by your formula.
Problem 2. In this problem we will do a similar but simpler construction as in the previous problem to find the area of a parallelogram. Draw a parallelogram and cut it out. Label the base b and height h, cut it into pieces, and reassemble the pieces to get a rectangle whose sides are b and h, respectively. Draw the original parallelogram and reassembled rectangle, and explain why your construction shows that the area of a parallelogram is equal to b × h. Describe why this formula is a special case of the formula you found in Problem 1.
