The area of a parallelogram is a key concept in geometry that tells us how much space this quadrilateral covers on a plane. To find it quickly and correctly, all you need are a few simple formulas. Let’s break them down and explore where they come from, how they work, and how to apply them in real-life examples.
How to Find the Area of a Parallelogram: Formula Using Side and Height
The simplest and most common way to calculate the area of a parallelogram is by multiplying the length of any side by the height drawn to that side (or its extension).
Imagine a parallelogram ABCD, where the height BM is drawn to side AD. The formula to calculate the area in this case is:
But where does this formula come from? Let’s explore it! By drawing another height, CN, to the extension of side AD, we create the quadrilateral MBСN. What shape is this? Yes, it’s a rectangle, because all its angles are right angles!
Now, let’s analyze this: the area of parallelogram ABCD is made up of two parts – the triangle ABM and the trapezoid MBCD. Meanwhile, the area of rectangle MBCN also divides into two parts – the trapezoid MBCD and triangle DCN.
Here’s the key point: the triangles ABM and DCN are exactly the same. Why? Because they have equal sides (AB=DC), and the angles between the sides and the heights match. This means that parallelogram ABCD and rectangle MBNC have the same area.
Calculating the area of a rectangle is simple: multiply its length by its width. In our case, that’s the sides MN (which equals AD) and height BM:
As you can see, this formula is simple, clear, and easy to prove.
Parallelogram Area Using Diagonals and the Angle Between Them: Another Convenient Method
Another helpful formula allows you to calculate the area of a parallelogram using its diagonals and the angle between them. Let’s dive into it.
Let’s return to parallelogram ABCD. Let’s label its diagonals AC and BD, which intersect at an angle α. The formula in this case looks like this:
How does this formula come about? It’s simple! When two diagonals are drawn, the parallelogram divides into four triangles. To find the area of each triangle, we use the standard formula for the area of a triangle based on two sides and the sine of the angle between them.
Once we add the areas of these four triangles, we get the total area of the parallelogram. A key point here: the two angles formed by the intersection of the diagonals are equal (α and β, with sin(α)=sin(180°-α)=sin(β)). So, when we simplify the sum of the areas of the four triangles, we arrive at the formula:
That’s it! While the formula might seem complicated at first, it’s actually quite simple and easy to memorize.
Note: If we denote the side of the parallelogram as a, the height to it as h, and the diagonals as d1 and d2, both area formulas can be written in the general form:
Learning to Apply the Formulas: Area of a Parallelogram in Practice
Now that we’ve gone through the formulas, it’s time to work through some examples. Let’s practice and see how quickly and easily we can calculate the area of a parallelogram. Each example includes a solution, but try to solve them on your own first, then check your answers.
Example 1: The Parallelogram Has a Base of 12 cm and a Height of 11 cm. What Is the Area of the Parallelogram?
The base and height of the parallelogram are given as 12 cm and 11 cm, respectively. Using the area formula, we get:
So, the area of a parallelogram is 132 cm2.
Example 2: What Is the Area of a Parallelogram with a Base of 15 cm and a Height of 20 cm?
In this case, the base and height of the parallelogram are 15 cm and 20 cm. Substituting these values into the area formula:
Thus, the area of a parallelogram is 300 cm2.
Example 3: The Area of the Parallelogram Is 36 cm2, and the Lengths of Its Heights Are 3 cm and 4 cm. Find the Perimeter of the Parallelogram
We can find the side lengths using the area and heights:
Now, we can find the perimeter: P=2⋅(AD+DC)=2⋅(12+9)=2⋅21=42 cm.
Example 4: The Diagonals of the Parallelogram Are 8 cm and 5 cm. The Angle Between Them Is 30 Degrees. Find the Area of the Parallelogram
We’ll use the formula for the area based on diagonals and the sine of the angle between them:
Therefore, the area of a parallelogram is 10 cm2.
Example 5: Find the Area of a Parallelogram with Sides Measuring 6 cm and 7 cm, and Angle ∠A=30°
First, drop a perpendicular from point B to side AD, which will intersect at point K (BK is the height).
Next, using the right triangle ABK, we find the length of the perpendicular (the leg opposite the 30-degree angle is half the hypotenuse):
Now, applying the formula for the area of a parallelogram:
So, the area of a parallelogram is 21 cm2.
What’s Next? Additional Materials to Expand Your Knowledge
Now that you understand how to calculate the area of a parallelogram in different ways, it’s time to explore other aspects of this shape. Have you ever wondered why diagonals are so important or how angles influence the shape of a parallelogram? To dive deeper, check out these additional resources:
- What is a Parallelogram and Its Properties? – Learn about the different types of parallelograms, their properties, and how to use them in calculations.
- Diagonal of a Parallelogram – Curious about how to find the length of a parallelogram’s diagonal? This resource includes formulas and step-by-step examples.
- Perimeter of a Parallelogram – Not sure how to calculate the perimeter correctly? This guide provides helpful tips and solutions.
Understanding these properties will help you approach parallelogram-related problems from different angles and solve even the most challenging problems. Keep expanding your mathematical knowledge!
Як Автоматизувати Обчислення: Блок-Схема для Вашого Коду
Want to combine geometry with programming? It’s a great way to reinforce what you’ve learned! Creating a simple program to calculate the area of a parallelogram isn’t as hard as it sounds. You just need to understand the formula and have basic programming skills. To get started, use the visual flowchart we’ve provided. It outlines the steps, the variables to use, and how to display results. Use it as a guide, experiment with your own code, and have fun building your program!
