Area of Parallelograms

Recap on parallelograms

Before we get into our main subject at hand, let us conduct a quick review on parallelograms to ease ourselves into this topic.

As the name implies, a parallelogram has parallel sides. Thus, we can define a parallelogram as below.

The following figure describes a parallelogram with sides, AB, BD, CD and AC.

Parallelogram illustration, StudySmarter Originals

Properties of parallelograms

We shall return to our parallelogram ABCD above. Let us look at some properties that distinguish this shape.

• The opposite sides of ABCD are parallel. In this case, AB is parallel to CD and AC is parallel to BD. We write this as AB // CD and AC // BD,

• The opposite angles of ABCD are equal. Here, ∠CAB = ∠CDB and ∠ACD = ∠ABD,

• The diagonals of a parallelogram bisect each other at a point, say M. Then, AM = MD and BM = MC. This is shown below,

Property of a parallelogram , StudySmarter Originals

• Each diagonal of a parallelogram divides the parallelogram into two congruent triangles. Triangle CAB is congruent to triangle CDB and triangle ACD is congruent to triangle ABD.

Types of parallelograms

There are three types of parallelograms we must consider throughout this syllabus, namely

1. Rectangle

2. Square

3. Rhombus

Each of these parallelograms has its distinct features that differentiate them from one another. A more detailed explanation of parallelograms can be found here, Parallelograms.

Area of parallelogram definition

In the diagram above, the total area enclosed by ABCD is the area of the parallelogram ABCD.

Area of Parallelogram Formula

Referring to our initial parallelogram ABCD, we shall add two new components to this figure called b and h. This is displayed in the diagram below.

A parallelogram with base b and height h, Study Smarter Originals

The variable b is called the base of the parallelogram. Either of the long sides of ABCD can be used as the base. For the diagram above, b can be either AB or CD. Here, here we have taken b = AB.

The variable h is called the height of the parallelogram. This may also be referred to as the altitude. The altitude is the line segment perpendicular to a pair of adjacent sides of the parallelogram with one endpoint on one side and the other endpoint on the other side.

Now that we have defined our variables b and h, we can thus present the area of a parallelogram as follows.

Area of parallelogram examples

With that in mind, let us now observe the following worked examples that make use of this formula.

Finding the Area of a Parallelogram from a Rectangle

Suppose we want to find the area of a parallelogram where the height (or altitude) is unknown. Instead, we are given the lengths of two sides of the parallelogram, namely the lengths of AB and AC.

Let us try looking at this scenario graphically. Referring back to our initial parallelogram ABCD, let us draw two altitudes for each pair of adjacent sides, AC and AB as well as CD and BD.

Area of a Parallelogram from a Rectangle, StudySmarter Originals

We thus obtain two new points on this parallelogram, namely S and T. Now observe the shape formed by BTCS. Does this look familiar to you? That's right! It's a rectangle, which is also a type of parallelogram. We now need to find a way to obtain the lengths of either CS or BT in order for us to deduce the height of this parallelogram.

Notice that from the construction of these two line segments, we have obtained a pair of right-angle triangles, CAS and BDT. Since CS = BT, it suffices for us to only calculate one of them. Let us take a look at triangle CAS.

Triangle CAS, StudySmarter Originals

For simplicity, we shall denote the following sides as: x = AS, y = CS and z = AC. Since this is a right-angle triangle, we can use Pythagoras' theorem to obtain the length of CS, which is the height of the parallelogram ABCD. Given the lengths of AS and AC, we have

x² + y² = z²

Rearranging this and applying the square root, we obtain

$y=\sqrt{z^2-x^2}$

As we have now found the length of CS, we can carry on finding the area of parallelogram ABCD by the formula given. We shall take the base as the length of AB. Thus, the area of ABCD is

$Are{a}_{ABCD}=AB\times CS$

Let us show this with an example.

Parallelogram Inscribed in a Rectangle Example

In this example, we shall look at a case where a parallelogram is inscribed inside a rectangle. We want to identify the area inside the rectangle that is not occupied by the parallelogram.

A Special Case: Area of the Rhombus

The rhombus is a special type of quadrilateral that in fact has its own formula for calculating its area. It is sometimes referred to as equilateral quadrilateral. Let us recall the definition of a rhombus.

We shall now consider the rhombus below. Two diagonals, AD (light blue line) and BC (dark blue line) are constructed on this parallelogram. The diagonals have lengths d₁ and d₂, respectively.

Area of a rhombus, StudySmarterOriginals

Area of a Rhombus

The area of the rhombus is given by the formula,

$A=\mathit{}\frac{\mathit{1}}{\mathit{2}}{d}_{\mathit{1}}{d}_{\mathit{2}}$

where A = area, d₁ = length of diagonal AD and d₂ = length of diagonal BC.

Example of the Area of a Rhombus

Here is an example involving the area of a rhombus formula.

• The formula for the area of a rhombus can also be used to find the area of a kite in a similar way.

We shall end this article with a final example involving the area of a parallelogram, or more specifically a kite.

Real-world Example of the Area of a Parallelogram

We shall now return to our example at the beginning of this article. As we now have a basic formula for calculating the area of a parallelogram, we can thus use it to find the area of our kite.