Welcome to the first lesson of your course on finding the area of geometric shapes! Building on what you already know about rectangles and triangles, we will start exploring some new shapes from your 8th-grade syllabus. Your goal is to become confident in solving area problems, and a great first step is mastering the parallelogram.
In this lesson, you will learn how to calculate the area of a parallelogram. We will cover its definition, understand why its area formula works, and practice applying it to solve problems. This will be the first formula for your collection and will form a strong foundation for the topics to come.
1. What is a Parallelogram?
A parallelogram is a flat, four-sided shape where both pairs of opposite sides are parallel and equal in length. Think of it as a rectangle that has been pushed over or slanted.
To find the area of a parallelogram, we need two key measurements: the base and the perpendicular height.
- The base can be any of its four sides. We usually choose the side that is at the bottom.
- The perpendicular height is the shortest distance from the base to the opposite side. It must form a right angle (90°) with the base.
This distinction is crucial: the height is not the length of the slanted side. It's the straight-up height.
The formula for the area of a parallelogram is the product of its base (b) and its perpendicular height (h). The example shows a parallelogram with a base of 8m and a perpendicular height of 6m, giving an area of 48 square meters.
Let's look at why this formula works. It's actually a clever trick that relates back to the area of a rectangle, which you already know.
2. From Parallelogram to Rectangle
Imagine you slice off a small triangle from one side of a parallelogram and move it to the other side. What shape do you get? A rectangle!
The following video provides a great visual demonstration of this transformation. Watching this will help you understand intuitively why the area formula for a parallelogram is so similar to that of a rectangle.
Find the Perimeter and Area of Parallelograms!
This video, "Find the Perimeter and Area of Parallelograms!" from the Quoc Dat Phung channel, visually shows how a parallelogram can be rearranged into a rectangle without changing its area.
Please watch the section from this clip. Pay close attention to how the triangular piece is moved to form a rectangle. Notice that the base of the parallelogram becomes the base (or length) of the rectangle, and the perpendicular height of the parallelogram becomes the height (or width) of the rectangle.
Because of this transformation, the area doesn't change. The area of the new rectangle is its base times its height. Therefore, the area of the original parallelogram is also its base times its perpendicular height.
3. Applying the Formula
Now that you understand the formula, let's see it in action. A common challenge is correctly identifying the base and the corresponding height, especially when extra information is given.
To avoid a common mistake, let's clarify the difference between the perpendicular height and the slant height.
Area of a Parallelogram - Math Steps, Examples & Questions
This section from Third Space Learning explains some of the most common errors made when calculating the area of a parallelogram.
Please read the section titled Easy mistakes to make. Focus on the third point, which highlights the error of using the diagonal (or slant) side length instead of the perpendicular height.
A parallelogram has two pairs of parallel sides, which means you can choose either pair as the base. The key is that the height you use must be perpendicular to the base you've chosen. The image below illustrates the different valid combinations of base and height.
This image shows four examples of a parallelogram. In each case, the area is calculated as base (b) times height (h). Notice that the height is always perpendicular to the chosen base, and sometimes the height must be drawn outside the shape to meet the extended base at a right angle.
Let's watch a few worked examples to solidify your understanding.
Find the Perimeter and Area of Parallelograms!
We'll return to the same video from Quoc Dat Phung to see two examples of calculating the area.
First, watch the example from the first problem. Then, continue with the example of the rotated parallelogram. Notice how in both cases, the formula base × height is applied directly.
4. Working Backwards: Finding a Missing Dimension
Sometimes, a problem might give you the area and one dimension (like the base) and ask you to find the missing dimension (the height). This is a good test of your understanding of the formula. It just requires a simple algebraic rearrangement.
If , then:
Let's see an example of this.
Find the Perimeter and Area of Parallelograms!
The next part of the video demonstrates how to find the height when the area and base are known.
Watch the final example from this clip. The presenter uses the area formula and divides the given area by the base to find the unknown height.
5. Your Formula Collection
As you requested, we will be collecting all the important formulas as we go. Here is a handy table that includes the parallelogram formula alongside other shapes you may be familiar with. You can start your formula sheet with this.
Area of Parallelogram: Definition, Formula And Worksheets
This resource from Think Academy provides a useful summary table of area formulas.
Review the section Wrapping Up. The table neatly organizes the area formulas for a square, rectangle, parallelogram, and trapezoid.
Conclusion
In this lesson, we have explored the parallelogram and its area. You've learned how to apply the formula and, just as importantly, why it works by relating it back to the area of a rectangle.
Here are the key takeaways:
- A parallelogram is a quadrilateral with two pairs of parallel sides.
- The area of a parallelogram is calculated using the formula: Area = base × perpendicular height.
- It is crucial to use the perpendicular height, not the length of the slanted side.
- The formula can be rearranged to find a missing base or height if the area is known.
In our next lesson, we will move on to another special quadrilateral: the rhombus. You'll see that a rhombus is just a special type of parallelogram, so the concepts you've learned today will be directly applicable.