Caculating Area: Parallelograms
Calculating the area of parallelograms is a fundamental concept in geometry, dating back to ancient civilizations. A parallelogram is defined as a four-sided polygon with opposite sides that are both equal and parallel. The area of a parallelogram can be determined using the formula A = bh, where "b" represents the length of the base and "h" is the height, the shortest distance from the base to the opposite side. This formula can be visually understood by either transforming the parallelogram into a rectangle or by dividing it into two congruent triangles.
The principles governing parallelograms also extend to other geometric shapes, such as trapezoids, kites, and rhombi, each with their own area calculation methods. For instance, while a trapezoid has only one pair of parallel sides, its area can be derived using the average length of its parallel sides. Notably, the properties of parallelograms, including congruent opposite sides and angles, are foundational in understanding more complex shapes like rectangles and squares.
An important relationship known as the Parallelogram Law illustrates that the sum of the squares of the four sides of a parallelogram equals the sum of the squares of its diagonals, highlighting the interconnectedness of these geometric figures. Overall, the study of parallelograms offers valuable insights and applications in various fields, including physics and engineering.
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Caculating Area: Parallelograms
One of the oldest areas of mathematics involves determining the area inside certain shapes. The parallelogram—a four-sided polygon with both pairs of opposite sides parallel, possesses some particularly interesting geometric properties. The name "parallelogram" means "bounded by parallel lines." Historical descriptions of the parallelogram for both mathematical and practical purposes can be traced back as far as ancient Babylonia. Even during that time period, a parallelogram’s area was known to be the product of the length of the base (any of the four sides) and the height (the shortest distance from the base to the side opposite the base). Two common ways to derive this formula are by equating the parallelogram with a rectangle (Figure 1) or by dividing a parallelogram into two congruent triangles (Figure 2).
Using the formula of A = bh for a parallelogram’s area, ancient societies were able to calculate the area of circles and construct special arches. But parallelograms are just as important today, with applications in areas such as physics, engineering, and industrial design.
Area of Trapezoids, Kites, Rhombi, and Rectangles
By slightly modifying the definition of a parallelogram, it is possible to derive many other interesting shapes. For example, a quadrilateral with only one pair of opposite sides parallel is known as a trapezoid. Because a trapezoid is not a parallelogram, the A = bh formula does not apply directly. However, by letting b represent the average length of a trapezoid’s parallel sides, the formula does hold. A simple geometric representation is shown in Figure 3.
One interesting property of a parallelogram is that opposite sides are congruent. It is also possible to construct a quadrilateral with two pairs of congruent sides, but with the congruent sides adjacent to each other. Such a figure is known as a kite. See Figure 4. The area of a kite is one half the product of the lengths of its diagonals. By constructing a rectangle with a base and height equal to the diagonals of a kite, one can see that the area of the kite is half that of the rectangle.
In the special case where all four sides of a quadrilateral area are congruent, the resulting figure is both a parallelogram and a kite. This special figure is known as a rhombus. Because a rhombus is a parallelogram and a kite, its area can be calculated using either formula.
Not only are opposite sides congruent in a parallelogram, but opposite angles are also congruent. A parallelogram with all four angles congruent is a rectangle. A rectangle with all four sides congruent is a square. For this reason, any properties that hold for parallelograms also hold for rectangles and squares, although the converse is not true (properties of rectangles or squares are not necessarily true for all parallelograms).
Parallelogram Law
Suppose we were to draw a rectangle with adjacent sides labeled a and b, and then label the diagonal that connects them as c (Figure 5). The Pythagorean theorem states that a2 + b2 = c2. Because this same relationship clearly is true for the other two sides (d and e) as well, and because the two diagonals (c and f) of a rectangle are of equal length, we can say that a2 + b2 + d2 + e2 = c2 + f2. Much more surprising is that this relationship between the sides and diagonals holds for all parallelograms. This relationship, known as the Parallelogram Law, states that the sum of the squares of the four side lengths of a parallelogram equals the sum of the squares of the two diagonal lengths.
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