How To Get Area Of Parallelogram

The area of a parallelogram, a fundamental concept in geometry, quantifies the two-dimensional space enclosed within its boundaries. Parallelograms, with their unique properties of having two pairs of parallel sides, are frequently encountered in various fields, from architecture and engineering to everyday spatial reasoning. Understanding how to accurately calculate their area is thus crucial for a wide range of applications.

Understanding Parallelograms: A Foundation

Before diving into the methods for calculating the area of a parallelogram, it's important to solidify our understanding of what defines a parallelogram. A parallelogram is a quadrilateral (a four-sided polygon) with two pairs of parallel sides. Key characteristics include:

• Opposite sides are parallel and equal in length: This is the defining feature of a parallelogram.
• Opposite angles are equal: Angles that are directly across from each other within the parallelogram are congruent.
• Consecutive angles are supplementary: Angles that are next to each other add up to 180 degrees.
• Diagonals bisect each other: The lines drawn from one corner to the opposite corner intersect at a point that divides each diagonal into two equal segments.

Familiarizing yourself with these properties will aid in visualizing and applying the area formulas we will explore.

Method 1: Base and Height – The Most Common Approach

The most common and straightforward method for calculating the area of a parallelogram utilizes its base and height.

Defining Base and Height:

• Base (b): Any side of the parallelogram can be chosen as the base. It is often the side that is lying horizontally.
• Height (h): The height is the perpendicular distance from the base to its opposite side. Perpendicular means that the height forms a right angle (90 degrees) with the base. It's crucial to remember that the height is not the length of the adjacent side unless that side is perpendicular to the base (making it a rectangle or square, special cases of parallelograms).

The Formula:

The area (A) of a parallelogram is calculated as:

A = b * h

Where:

• A = Area
• b = Length of the base
• h = Height

Step-by-Step Calculation:

1. Identify the base: Choose one side of the parallelogram as the base. Measure its length.
2. Determine the height: Find the perpendicular distance from the base to the opposite side. This might be given directly in a problem or require measurement.
3. Apply the formula: Multiply the length of the base by the height. The result is the area of the parallelogram.
4. Include units: Remember to express the area in square units (e.g., cm², m², in², ft²), as area represents a two-dimensional space.

Example 1:

Imagine a parallelogram with a base of 10 cm and a height of 5 cm. Using the formula, the area would be:

A = 10 cm * 5 cm = 50 cm²

Example 2:

Suppose a parallelogram has a base of 8 inches and a height of 6 inches. The area is:

A = 8 in * 6 in = 48 in²

Why does this formula work?

This formula works because a parallelogram can be transformed into a rectangle with the same base and height. Imagine cutting off a right-angled triangle from one end of the parallelogram and attaching it to the other end. This creates a rectangle with the same base and height as the original parallelogram. Since the area of a rectangle is base times height, the area of the parallelogram is also base times height. This visualization provides an intuitive understanding of the formula's validity.

Method 2: Using Trigonometry (When Height is Unknown)

Sometimes, the height of the parallelogram isn't directly provided. Instead, you might be given the length of one of the sides adjacent to the base and the angle between that side and the base. In such cases, trigonometry can be employed to find the height.

Understanding the Trigonometric Approach:

This method utilizes the sine function, a fundamental concept in trigonometry. The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle (in this case, the height of the parallelogram) to the length of the hypotenuse (in this case, the side adjacent to the base).

The Formula:

The area (A) of a parallelogram, when given the base (b), the adjacent side (a), and the angle (θ) between them, is calculated as:

A = b * a * sin(θ)

Where:

• A = Area
• b = Length of the base
• a = Length of the adjacent side
• θ = Angle between the base and the adjacent side (in degrees or radians)
• sin(θ) = The sine of the angle θ

Step-by-Step Calculation:

1. Identify the base (b) and adjacent side (a): Determine the lengths of the base and the side adjacent to the base.
2. Determine the angle (θ): Find the angle between the base and the adjacent side.
3. Calculate the sine of the angle: Use a calculator or trigonometric table to find the sine of the angle θ. Make sure your calculator is in the correct mode (degrees or radians) depending on the units of the angle.
4. Apply the formula: Multiply the base (b) by the adjacent side (a) and the sine of the angle (sin(θ)). The result is the area of the parallelogram.
5. Include units: Express the area in square units.

Example 1:

Consider a parallelogram with a base of 12 meters, an adjacent side of 7 meters, and the angle between them is 60 degrees.

1. b = 12 m
2. a = 7 m
3. θ = 60°
4. sin(60°) ≈ 0.866 (using a calculator)
5. A = 12 m * 7 m * 0.866 ≈ 72.74 m²

Therefore, the area of the parallelogram is approximately 72.74 square meters.

Example 2:

Suppose a parallelogram has a base of 5 inches, an adjacent side of 4 inches, and an angle of 30 degrees between them.

1. b = 5 in
2. a = 4 in
3. θ = 30°
4. sin(30°) = 0.5
5. A = 5 in * 4 in * 0.5 = 10 in²

Therefore, the area of the parallelogram is 10 square inches.

Derivation of the Formula:

This formula is derived from the basic area formula (A = b * h) and the definition of the sine function. In the right triangle formed by the height, the adjacent side, and a portion of the base, we have:

sin(θ) = h / a

Solving for h, we get:

h = a * sin(θ)

Substituting this value of h into the basic area formula, we obtain:

A = b * (a * sin(θ)) = b * a * sin(θ)

This demonstrates how the trigonometric approach is simply a variation of the base-height method, adapted for situations where the height is not directly given.

Method 3: Using Diagonals (and the Angle Between Them)

Another way to calculate the area of a parallelogram is by using the lengths of its diagonals and the angle between them. This method is particularly useful when the base and height are not readily available, but the diagonals and their intersection angle are known.

Understanding Diagonals and Their Properties:

As mentioned earlier, diagonals are lines drawn from one corner of the parallelogram to the opposite corner. In a parallelogram, the diagonals bisect each other, meaning they intersect at their midpoints.

The Formula:

The area (A) of a parallelogram, when given the lengths of its diagonals (d1 and d2) and the angle (φ) between them, is calculated as:

A = (1/2) * d1 * d2 * sin(φ)

Where:

• A = Area
• d1 = Length of the first diagonal
• d2 = Length of the second diagonal
• φ = Angle between the diagonals (in degrees or radians)
• sin(φ) = The sine of the angle φ

Step-by-Step Calculation:

1. Identify the diagonals (d1 and d2): Determine the lengths of both diagonals of the parallelogram.
2. Determine the angle (φ): Find the angle between the diagonals at their point of intersection. Remember that the diagonals bisect each other.
3. Calculate the sine of the angle: Use a calculator or trigonometric table to find the sine of the angle φ. Ensure your calculator is in the correct mode (degrees or radians).
4. Apply the formula: Multiply one-half (1/2) by the length of the first diagonal (d1), the length of the second diagonal (d2), and the sine of the angle (sin(φ)).
5. Include units: Express the area in square units.

Example 1:

Consider a parallelogram with diagonals of length 8 cm and 10 cm, and the angle between them is 75 degrees.

1. d1 = 8 cm
2. d2 = 10 cm
3. φ = 75°
4. sin(75°) ≈ 0.966 (using a calculator)
5. A = (1/2) * 8 cm * 10 cm * 0.966 ≈ 38.64 cm²

Therefore, the area of the parallelogram is approximately 38.64 square centimeters.

Example 2:

Suppose a parallelogram has diagonals of 6 meters and 14 meters, and the angle between them is 45 degrees.

1. d1 = 6 m
2. d2 = 14 m
3. φ = 45°
4. sin(45°) ≈ 0.707
5. A = (1/2) * 6 m * 14 m * 0.707 ≈ 29.7 m²

Therefore, the area of the parallelogram is approximately 29.7 square meters.

Why does this formula work?

The derivation of this formula involves dividing the parallelogram into four triangles by its diagonals. The area of each triangle can be expressed using the formula (1/2) * side1 * side2 * sin(angle). By summing the areas of these four triangles and utilizing the properties of parallelograms and the sine function, the formula A = (1/2) * d1 * d2 * sin(φ) can be derived. A detailed geometric proof involves trigonometric identities and can be found in advanced geometry texts. Essentially, this formula cleverly uses the diagonals and their angle to indirectly calculate the base and height, leading to the area.

Special Cases: Rectangles and Squares

It's important to remember that rectangles and squares are special cases of parallelograms. Therefore, the area formulas discussed above apply to them as well. However, due to their specific properties, simpler formulas can be used.

• Rectangle: A rectangle is a parallelogram with four right angles. Its area is simply the product of its length (l) and width (w): A = l * w. This is equivalent to the base-height formula for parallelograms since the length and width are perpendicular to each other.
• Square: A square is a rectangle with all sides equal in length. If the side length of a square is 's', then its area is A = s². This is a special case of the rectangle area formula where length and width are equal.

When dealing with rectangles and squares, using these specific formulas is generally more efficient. However, understanding that they are derived from the general parallelogram formulas reinforces the interconnectedness of geometric concepts.

Practical Applications

Calculating the area of parallelograms has numerous practical applications in various fields:

• Architecture and Construction: Determining the amount of material needed for flooring, roofing, or wall coverings when dealing with parallelogram-shaped surfaces.
• Engineering: Calculating the surface area of components in machines or structures with parallelogram cross-sections.
• Land Surveying: Determining the area of land plots that are shaped like parallelograms.
• Art and Design: Calculating the area of canvases or other art materials with parallelogram shapes.
• Everyday Life: Estimating the amount of fabric needed to make a parallelogram-shaped banner or the amount of paint needed to cover a parallelogram-shaped wall.

Understanding these applications highlights the practical relevance of mastering the area calculation methods discussed.

Common Mistakes to Avoid

When calculating the area of a parallelogram, it's crucial to avoid common mistakes that can lead to incorrect results:

• Confusing the height with the adjacent side: The height must be perpendicular to the base. Using the length of the adjacent side instead of the perpendicular height will result in an inaccurate area.
• Using incorrect units: Ensure that all measurements are in the same units before performing calculations. If the base is in centimeters and the height is in meters, convert one of them before multiplying. Remember to express the final area in square units.
• Incorrectly applying trigonometric functions: Make sure your calculator is in the correct mode (degrees or radians) when using the sine function. Double-check the angle you are using in the formula.
• Forgetting the (1/2) factor when using diagonals: The formula involving diagonals includes a factor of (1/2). Forgetting this factor will lead to an area that is twice the correct value.
• Assuming all quadrilaterals are parallelograms: Only apply the parallelogram area formulas to shapes that actually meet the definition of a parallelogram (two pairs of parallel sides).

By being mindful of these common pitfalls, you can significantly improve the accuracy of your area calculations.

Conclusion

Calculating the area of a parallelogram is a fundamental skill in geometry with diverse applications. This article has explored three distinct methods for finding the area: using the base and height, employing trigonometry when the height is unknown, and utilizing the lengths of the diagonals and the angle between them. Each method offers a unique approach and is suitable for different scenarios depending on the information provided. Understanding the underlying principles and derivations of these formulas provides a deeper appreciation for their validity and applicability. Furthermore, recognizing special cases like rectangles and squares and avoiding common mistakes will ensure accurate and confident area calculations. Mastering these techniques empowers you to solve a wide range of problems involving parallelograms in various fields, from architecture and engineering to everyday spatial reasoning.