A trapezoid is a special type of quadrilateral that has at least one pair of parallel sides. These parallel sides are called the bases of the trapezoid, with the longer one typically called the lower base and the shorter one the upper base. The non-parallel sides are called the legs.
The area formula for a trapezoid is: Area equals the sum of the upper base and lower base, multiplied by the height, then divided by 2. In mathematical notation, this is written as A equals a plus b times h, all divided by 2, where a is the upper base, b is the lower base, and h is the height.
The first method to derive the trapezoid area formula involves taking two identical trapezoids. We rotate one trapezoid 180 degrees and combine them together. This creates a parallelogram whose base equals the sum of the upper and lower bases, and whose height equals the trapezoid's height. Since the parallelogram area is base times height, and it contains two trapezoids, one trapezoid's area is half the parallelogram area.
The second method involves drawing a diagonal line to divide the trapezoid into two triangles. The first triangle has base equal to the lower base b and height h, so its area is b times h divided by 2. The second triangle has base equal to the upper base a and the same height h, so its area is a times h divided by 2. The total trapezoid area is the sum of these two triangle areas, which equals a plus b times h, all divided by 2.
In conclusion, both derivation methods lead us to the same trapezoid area formula: Area equals the sum of the upper base and lower base, multiplied by the height, then divided by 2. This fundamental formula is essential in geometry and has practical applications in calculating land areas, designing engineering structures, and solving real-world problems involving trapezoidal shapes.