Area of Parallelogram: Formula and Applications

Area of Parallelogram

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. The area of a parallelogram is the region it encloses and can be calculated using its base and height. In this article, we will explore the relationship between the base and height of a parallelogram, the formula for finding the area, comparisons to other polygons, real-world applications, and various proofs and derivations.

Relationship to Base and Height

The base and height of a parallelogram are directly proportional to the area of the parallelogram. The height is the perpendicular distance between the base and the opposite side, and it is also known as the altitude or the vertical height. The base is one of the parallel sides of the parallelogram. The area of a parallelogram is given by the formula:

$$A = Bh$$

where:

• A is the area of the parallelogram
• B is the base of the parallelogram
• h is the height of the parallelogram

Formula for Finding Area

The formula for finding the area of a parallelogram is:

$$A = Bh$$

where:

• A is the area of the parallelogram
• B is the base of the parallelogram
• h is the height of the parallelogram

Comparison to Other Polygons

Parallelograms are similar to rectangles in that they both have parallel sides. However, parallelograms can have different base lengths and heights, while rectangles have equal base lengths and heights. The area of a rectangle is given by the formula:

$$A = Bh$$

where:

• A is the area of the rectangle
• B is the base of the rectangle
• h is the height of the rectangle

Real-World Applications

Parallelograms have various real-world applications, such as in construction. For example, the area of a piece of roofing material can be calculated as a parallelogram, with the base as the length of the roof and the height as the thickness of the roofing material. This allows for accurate measurement and estimation of the amount of roofing material needed for a project.

Proofs and Derivations

There are several ways to prove and derive the formula for the area of a parallelogram. One common method is to use the concept of a rectangle, which is a special case of a parallelogram where both base lengths are equal. The area of a rectangle can be calculated using the formula:

$$A = Bh$$

where:

• A is the area of the rectangle
• B is the base of the rectangle
• h is the height of the rectangle

Now, consider a parallelogram ABCD with base AD and height BC. To find the area of the parallelogram, we can divide it into two rectangles:

1. Rectangle ABCF with base AC and height BC
2. Rectangle BCFD with base CD and height BC

The area of rectangle ABCF is given by:

$$A_{ABCF} = AC \times BC$$

The area of rectangle BCFD is given by:

$$A_{BCFD} = CD \times BC$$

The total area of the parallelogram is the sum of the areas of these rectangles:

$$A_{ABCD} = A_{ABCF} + A_{BCFD}$$

Substituting the respective base and height values, we get:

$$A_{ABCD} = (AD + CD) \times BC$$

Since AD and CD are equal (as they are both base lengths of the parallelogram), we can simplify this expression to:

$$A_{ABCD} = 2BD \times BC$$

Now, if we let BD represent the base B and BC represent the height h of the parallelogram, we have:

$$A_{ABCD} = 2B \times h$$

Dividing both sides by 2, we get the final formula for the area of a parallelogram:

$$A_{ABCD} = Bh$$

This formula can be used to find the area of any parallelogram, given its base and height.