Proving that the Area of the Trapezoid is Equal to the Area of the Parallelogram Formed from It

Abstract

INTRODUCTION

Two of the most frequently used geometric shapes are the trapezoids and parallelograms. A trapezoid is a 4-sided figure with one pair of parallel sides. Finding the area equals the sum of the bases, b1 and b2 is divided into two and is multiplied by height h. The perpendicular distance between the bases is known as the height. The sides that are not parallel are called legs, and a line from the midpoint of one leg to the midpoint of the other is called the median. A parallelogram is also a four-sided polygon in which each side is equal in length to its opposite side, and are parallel to each other. The area is the product of one side, taken as a base, times the shortest distance to the opposite side. These are two different figures. Thus, they have different formulas in finding their area. The end in mind of the proponents is to prove that the area of the trapezoid is equal to the area of the parallelogram formed from it.

METHODS

Three different trapezoids of different measures are to be used for the three conjectures and each trapezoid will be cut in the median in order to form a parallelogram. Procedure: (1) Draw a trapezoid of any shape or size on a piece of colored paper. (2) Cut out the trapezoid. Label the bases and height. (3) Measure the lengths of b1 and b2. Measure the height. Record the measurements. (4) Fold b1 onto b2. Unfold. (5) Cut the trapezoid on the fold line. Then form a parallelogram. (6) Measure the height and base of the parallelogram formed from the cut-out trapezoid. Record the measurement. (7) Compute the Area of the Trapezoid and Area of the Parallelogram.

RESULTS

Following the procedure, three geometric shapes have been formed-three parallelograms were created from the trapezoids by folding b1 onto b2 and cut out the trapezoid in the folded portion. The bases and height of the parallelograms were being measured and the area had been computed. Upon looking at the data collected, we can observe that the Area computed using the Trapezoid has the same value to the Area computed using the Parallelogram.

DISCUSSIONS

After a thorough investigation using the three trials of different measures of a trapezoid, we can therefore conclude and have proven that the area of the trapezoid is equal to the area of the parallelogram formed from it. This finding would enable us to solve directly the area of a parallelogram formed from a given trapezoid. When a particular trapezoid is given, just form a parallelogram out from it, measure the base and the height, we can already solve for the area.