Problem 55 - Thinking Mathematically

Question

Using the formula for the area of a rectangle, explain how the formula for the area of a parallelogram \((A=b h)\) is obtained.

Step-by-Step Solution

Answer

The formula for the area of a parallelogram can be derived from the formula for the area of a rectangle by transforming a parallelogram into a rectangle. The area is given by the formula \( A=bh \), where \( b \) is the base and \( h \) is the height of the parallelogram.

Step 1: Understand parallelogram and rectangle

A rectangle is a four-sided figure with all right angles. The area of a rectangle is calculated by the product of the length \( l \) and width \( w \), that is \( A=lw \). A parallelogram, on the other hand, is a quadrilateral in which both pairs of opposite sides are parallel. In a parallelogram, the base \( b \) is any of the sides and the height \( h \) is the perpendicular distance between the base and the opposite side.

Step 2: Manipulate a parallelogram into a rectangle

Draw a parallelogram. Cut a right triangle from one end of the parallelogram and then move it to the opposite side. Now it becomes a rectangle.

Step 3: Calculate the area using the rectangle's area

Now that the parallelogram has been reshaped into a rectangle, the area can be calculated using the formula for the rectangle's area, \( A=lw\). Because the length of the rectangle corresponds to the base of the original parallelogram \( b \), and the width corresponds to the height of the parallelogram \( h \), the formula becomes \( A=bh \).

Key Concepts

Parallelogram Properties

A parallelogram is a special type of quadrilateral with some intriguing properties that set it apart. Each pair of opposite sides is equal in length and parallel, which is a defining trait of parallelograms. Another important property is that the opposite angles are also equal, and consecutive angles are supplementary (add up to 180 degrees).

Understanding these properties is critical when studying geometric transformations and calculating areas. The area of a parallelogram is equal to the product of its base and the perpendicular height. This concept is fundamental because it underpins the relationship between parallelograms and other quadrilaterals, such as rectangles, in terms of area calculation.

Geometric Transformations

Geometric transformations are operations that alter the position, shape, or size of a figure in a mathematical space. Common transformations include translations, rotations, reflections, and dilations. However, the transformation we're focusing on here is the 'shear' transformation, which can turn a rectangle into a parallelogram and vice versa without changing the area.

This property of congruent areas despite shape changes is pivotal because it allows us to understand how a parallelogram can essentially have the same area formula as a rectangle. When you manipulate a parallelogram by shearing it, the base and height remain constant, and thus, so does the area.

Area Calculation

Calculating the area of a parallelogram involves a straightforward formula, where the area (\(A\)) is the product of the base (\(b\)) and the height (\(h\)). The base is any side length of the parallelogram, and the height is the perpendicular distance from that base to the opposite side.

The significance of this formula is its uniform application across varying types of parallelograms, whether they have right angles (like rectangles) or not. Understanding and correctly identifying the base and the height is crucial for accurate area calculations, making this concept an essential part of geometry education.

Rectangles

A rectangle can be thought of as a special case of a parallelogram where all angles are right angles (90 degrees). Due to this, both pairs of opposite sides are not only parallel but also of equal length. The area of a rectangle is given by the formula \(A = lw\), where \(l\) is the length and \(w\) is the width of the rectangle.

When the formula for the area of a rectangle is applied to a parallelogram, it provides insight into the underlying uniformity between different quadrilaterals. Rectangles help in visualizing area calculations for parallelograms, and in some cases, geometric transformations can convert a parallelogram into a rectangle to Simplify computation.