How to Find the Area of a Parallelogram: A Comprehensive Guide

Understanding how to calculate the area of different shapes is a fundamental skill in geometry and has practical applications in everyday life. Among quadrilaterals, the parallelogram holds a special place due to its unique properties and straightforward area calculation. This guide will walk you through everything you need to know about finding the area of a parallelogram, from basic formulas to more complex problems, ensuring you grasp the concept completely.

Understanding the Basics of a Parallelogram

Before diving into the area calculation, let’s define what a parallelogram is and its key characteristics. A parallelogram is a four-sided shape, also known as a quadrilateral, distinguished by having two pairs of parallel sides.

Key Properties of a Parallelogram:

• Parallel Sides: Opposite sides are parallel to each other.
• Equal Opposite Sides: Opposite sides are not only parallel but also equal in length (congruent).
• Equal Opposite Angles: Angles opposite each other within the parallelogram are equal (congruent).
• Angles Sum to 360°: The interior angles of any parallelogram always add up to 360 degrees.

These properties make parallelograms unique and influence how we calculate their area.

Defining Base and Height for Area Calculation

To find the area of a parallelogram, we need to understand two crucial measurements: the base and the perpendicular height.

• Base (b): The base of a parallelogram is any of its sides, usually considered to be the bottom side for simplicity.
• Perpendicular Height (h): The height is the perpendicular distance between the base and its opposite parallel side. It’s crucial to understand that the height is not usually the length of the slanted side of the parallelogram. The perpendicular height forms a right angle (90 degrees) with the base.

What is the area of a parallelogram?

It’s important to correctly identify the perpendicular height, especially when dealing with parallelograms where the side length is also given, as this can be misleading.

Method 1: Using the Formula to Calculate the Area of a Parallelogram

The most direct method to find the area of a parallelogram is by using a simple formula. This formula is derived from the relationship between parallelograms and rectangles. Imagine cutting a right-angled triangle from one side of a parallelogram and attaching it to the other side. This transforms the parallelogram into a rectangle with the same base and height. Since the area of a rectangle is base times height, the area of a parallelogram is calculated using the same principle.

The Formula:

Area of Parallelogram = base × perpendicular height
Area = b × h

Where:

• Area is the space enclosed within the parallelogram, measured in square units.
• b is the length of the base of the parallelogram.
• h is the perpendicular height of the parallelogram.

Step-by-step guide to find the area using the formula:

1. Identify the base (b) and the perpendicular height (h) of the parallelogram. These measurements will usually be provided in the problem or diagram. Remember to use the perpendicular height, not the slanted side length.
2. Write down the formula for the area of a parallelogram: Area = b × h
3. Substitute the values of the base and height into the formula. Replace ‘b’ and ‘h’ with their respective numerical values.
4. Calculate the area. Multiply the base by the perpendicular height to get the numerical value of the area.
5. Write down your final answer with the correct square units. Area is always measured in square units, such as square meters (m²), square centimeters (cm²), square feet (ft²), etc., depending on the units used for the base and height.

Example:

Let’s say a parallelogram has a base of 8 meters and a perpendicular height of 6 meters.

1. Base (b) = 8 m, Perpendicular height (h) = 6 m
2. Formula: Area = b × h
3. Substitute: Area = 8 m × 6 m
4. Calculate: Area = 48
5. Final Answer: Area = 48 m²

Therefore, the area of the parallelogram is 48 square meters.

Method 2: Decomposing and Rearranging a Parallelogram to Find the Area

Another way to understand and calculate the area of a parallelogram is through decomposition and rearrangement. This method visually demonstrates why the formula Area = base × height works.

Steps to decompose and find the area:

1. Decompose the parallelogram: Imagine drawing a perpendicular line from one of the vertices of the parallelogram to the base, effectively cutting off a right-angled triangle from one side. This decomposes the parallelogram into a right-angled triangle and a trapezoid (or a rectangle and another right-angled triangle).
2. Rearrange the parts: Mentally or physically move the right-angled triangle you cut off to the opposite side of the parallelogram. Attach it so that the slanted side of the triangle aligns with the slanted side of the remaining shape.
3. Form a rectangle: By moving the triangle, you will notice that the original parallelogram has been transformed into a perfect rectangle.
4. Identify the base and height of the rectangle: The base of this rectangle is the same as the base of the original parallelogram. The height of the rectangle is the perpendicular height of the original parallelogram.
5. Apply the area of a rectangle formula: Since you now have a rectangle, use the familiar formula for the area of a rectangle, which is Area = Base × Height. In this context, ‘Base’ and ‘Height’ refer to the base and perpendicular height of the original parallelogram.
6. Calculate and state the answer in square units.

Visual Example:

Imagine a parallelogram.

Image showing a parallelogram being decomposed by a vertical cut.

Cut off the triangular section.

Image showing the triangular section cut off.

Move and attach the triangle to the other side to form a rectangle.

Image showing the parallelogram transformed into a rectangle.

This visual method helps to solidify the understanding that the area of a parallelogram is indeed calculated in the same way as a rectangle – base times perpendicular height.

Area of a Parallelogram Examples and Practice Problems

Let’s work through some examples to apply these methods and tackle different scenarios you might encounter.

Example 1: Basic Area Calculation

Problem: Find the area of a parallelogram with a base of 10 meters and a perpendicular height of 6 meters using the decomposition method.

Solution:

1. Decompose and Rearrange: As shown in the visual method above, imagine transforming the parallelogram into a rectangle.
2. Identify Base and Height of Rectangle: The base of the rectangle is 10 m, and the height is 6 m.
3. Area of Rectangle Formula: Area = Base × Height
4. Substitute and Calculate: Area = 10 m × 6 m = 60
5. Final Answer: The area of the parallelogram is 60 m².

Example 2: Dealing with Extra Information

Problem: Calculate the area of the parallelogram below. Notice there’s an extra side length given.

Image of a parallelogram with base 8m, perpendicular height 6m, and diagonal side 6.3m.

Solution:

1. Identify Base and Perpendicular Height: From the diagram, the base is 8 m, and the perpendicular height is 6 m. The 6.3 m measurement is the length of the diagonal side and is not needed for area calculation.
2. Area Formula: Area = base × perpendicular height
3. Substitute and Calculate: Area = 8 m × 6 m = 48
4. Final Answer: The area is 48 m².

Key takeaway: Always use the perpendicular height, and disregard any diagonal side lengths when calculating the area of a parallelogram unless the problem specifically requires it for a different calculation.

Example 3: Unit Conversion

Problem: Find the area of a parallelogram with a base of 10 meters and a perpendicular height of 750 centimeters.

Solution:

1. Convert Units: Before calculating, ensure units are consistent. Let’s convert centimeters to meters. Since 100 cm = 1 m, 750 cm = 750 / 100 = 7.5 m.
2. Identify Base and Height in Meters: Base = 10 m, Height = 7.5 m.
3. Area Formula: Area = base × perpendicular height
4. Substitute and Calculate: Area = 10 m × 7.5 m = 75
5. Final Answer: The area is 75 m².

Remember: Always check and convert units to be the same before performing calculations to avoid errors.

Example 4: Finding Base Length Given Area

Problem: The area of a parallelogram is 242 cm², and its perpendicular height is 11 cm. Find the length of the base.

Solution:

1. Area Formula: Area = base × perpendicular height
2. Rearrange Formula to Solve for Base: Base = Area / perpendicular height
3. Substitute Values: Base = 242 cm² / 11 cm
4. Calculate: Base = 22 cm
5. Final Answer: The base length is 22 cm. Note that the unit is in cm, not cm², because we are finding a length, not an area.

Example 5: Real-World Application

Problem: A painter needs to paint a logo made of four identical parallelograms. Each parallelogram has a base of 4 meters and a perpendicular height of 2 meters. If one can of paint covers 2.5 m² and costs $1.20, how much will it cost to paint the entire logo?

Solution:

1. Area of One Parallelogram: Area = base × perpendicular height = 4 m × 2 m = 8 m²
2. Total Area of Logo (4 parallelograms): Total Area = 4 × 8 m² = 32 m²
3. Number of Paint Cans Needed: Cans = Total Area / Coverage per can = 32 m² / 2.5 m² per can = 12.8 cans
4. Round Up to Whole Cans: Since you can’t buy a fraction of a can, round up to 13 cans.
5. Total Cost: Total Cost = Number of Cans × Cost per can = 13 cans × $1.20/can = $15.60
6. Final Answer: It will cost $15.60 to paint the logo.

Common Mistakes to Avoid When Calculating Parallelogram Area

• Incorrect Units: Forgetting to use square units for area in the final answer. Always remember area is a two-dimensional measurement and requires square units.
• Unit Conversion Errors: Failing to convert all measurements to the same unit before calculation. Double-check units and convert if necessary.
• Confusing Height with Side Length: Using the slanted side length instead of the perpendicular height in the area formula. Always use the perpendicular distance between the base and its opposite side as the height.

Practice Questions

Test your understanding with these practice questions:

1. Find the area of a parallelogram with a base of 12 m and a perpendicular height of 5.5 m.
2. A parallelogram has a base of 7 m and a perpendicular height of 450 cm. Calculate its area in square meters.
3. A garden is shaped like a parallelogram with a square fountain inside. The parallelogram has a base of 8 m, a perpendicular height of 5 m, and the square fountain has sides of 2 m. What is the area of the garden excluding the fountain?
4. The area of a parallelogram is 216 cm², and its perpendicular height is 9 cm. What is the length of its base?
5. Calculate the total shaded area in the figure below, which combines a parallelogram and a rectangle.
   (Note: You would need to provide an image for question 5, similar to the original article’s question 5, for a complete practice)
6. A landscaper is designing a rectangular garden that includes a parallelogram-shaped patio and a parallelogram-shaped sandpit. The garden is 15m long and 7m wide. The patio is 5m base and 4m height, and the sandpit is 1.5m base and 4.5m height. What area of the garden will be covered in grass?

(Answers to questions 1, 2, 3, 4, and 6 are available in the original article’s practice questions section. Question 5 would require a similar diagram to question 5 in the original article)

Frequently Asked Questions (FAQs) About Parallelogram Area

How do you calculate the area of a parallelogram?
The area of a parallelogram is calculated by multiplying the length of its base by its perpendicular height. Ensure you are using the perpendicular height, which is the vertical distance between the base and the opposite side.

What is the formula for the area of a parallelogram?
The formula is: Area = base × perpendicular height, or Area = b × h.

Can the adjacent sides of a parallelogram be at right angles?
Yes, if the adjacent sides of a parallelogram are at right angles (90 degrees), it becomes a special type of parallelogram known as a rectangle. A rectangle is indeed a parallelogram where all angles are right angles.

Next Steps in Geometry

After mastering the area of a parallelogram, you can explore related topics in geometry, such as:

• Area of other quadrilaterals (rectangles, squares, trapezoids, rhombuses)
• Area of triangles
• Area of polygons
• Surface area and volume of 3D shapes

By building a strong foundation in calculating areas of 2D shapes, you’ll be well-prepared for more advanced geometrical concepts and real-world applications.

Need More Help?

If you’re still finding geometry challenging, remember that personalized math support can make a significant difference. Consider seeking help from math tutors or exploring online resources to further enhance your understanding and confidence in math.