Area of Quadrilateral

The area of a quadrilateral is the amount of region that is present inside it. Let us recall what is a quadrilateral. A quadrilateral is a closed shape that is bounded by four line segments. A quadrilateral can be regular or irregular. A regular quadrilateral is a quadrilateral in which all sides are of equal length. A quadrilateral that is not regular is called an irregular quadrilateral. There are 6 types of quadrilaterals.

• square
• rectangle
• parallelogram
• trapezoid
• rhombus
• kite

In this page, we will see how to find the area of a quadrilateral by splitting it as two triangles and how to find the area of a quadrilateral using its 4 sides. Also, we will learn the formulas to find the area of each of these different types of quadrilaterals.

The area of a quadrilateral is nothing but the region enclosed by the sides of the quadrilateral. It is measured in square units such as m², in², cm², etc. The process of finding the area of a quadrilateral depends upon its type and the information available about the quadrilateral. If the quadrilateral does not belong to one of the types that are mentioned above, then we can find its area either by dividing it into two triangles or by using the formula (which is called the Bretschneider′s formula) of finding the area of quadrilateral using four sides. Here you can see the formulas to find the area of a quadrilateral which does not belong to any of the standard types.

Let us learn more about these formulas in the upcoming sections.

Consider a quadrilateral ABCD in which the length of the diagonal BD is known to be 'd'. ABCD can be divided into two triangles by the diagonal BD. To find its area, we should be knowing the heights of the triangles ABD and BCD. Let us assume that the heights of the triangles BCD and ABD are given to be \(h_1\) and \(h_2\) respectively. We will find the area of the quadrilateral ABCD by adding the areas of the triangles BCD and ABD.

Here, the area of the triangle BCD = (1/2) × d × \(h_1\).

The area of the triangle ABD = (1/2) × d × \(h_2\).

From the above figure, the area of the quadrilateral ABCD = area of ΔBCD + area of ΔABD.

Thus, the area of the quadrilateral ABCD = (1/2) × d × \(h_1\) + (1/2) × d × \(h_2\) = (1/2) × d × (\(h_1+h_2\)).

Thus, the formula used to find the area of a quadrilateral when one of its diagonals and the heights of the triangles (formed by the given diagonal) are given is,

Area = (1/2) × Diagonal × (Sum of heights)

When the sides of a quadrilateral and two of its opposite angles are given, we can find its area using the Bretschneider′s formula. Let us consider a quadrilateral whose sides are a, b, c, and d, and two of its opposite angles are θ\(_1\) and θ\(_2\).

Then the area of the quadrilateral = \(\sqrt{(s-a)(s-b)(s-c)(s-d)-a b c d \cos ^{2} \frac{\theta}{2}}\), where

• s = semi-perimeter of the quadrilateral = (a + b + c + d)/2
• θ = θ\(_1\) + θ\(_2\)

By Heron's formula, the area of a triangle with 3 sides a, b, and c is \(\sqrt{s(s-a)(s-b)(s-c)}\), where 's' is the semi-perimeter of the triangle, i.e., s = (a + b + c)/2. To find the area of a quadrilateral using Heron's formula,

• Divide it into two triangles using a diagonal (Use the diagonal whose length is known).
• Apply Heron's formula for each of the triangles to find its area.
• Add the areas of two triangles which gives the area of the quadrilateral.

We already learned that there are 6 types of quadrilaterals, which are, square, rectangle, parallelogram, trapezoid, rhombus, and kite. We have a specific formula to find the area of each of these quadrilaterals. Let us see them.

The area of a quadrilateral can be calculated when the coordinates of its vertices are known. Let us consider a quadrilateral in coordinate plane as shown below,

In the quadrilateral given above, A(x\(_1\), y\(_1\)), B(x\(_2\), y\(_2\)), C(x\(_3\), y\(_3\)) and D(x\(_4\), y\(_4\)) are the vertices.

To find area of the quadrilateral ABCD, we take the vertices A(x\(_1\), y\(_1\)), B(x\(_2\), y\(_2\)), C(x\(_3\), y\(_3\)) and D(x\(_4\), y\(_4\)) of the quadrilateral ABCD and write them as shown below,

Add the diagonal products x\(_1\)y\(_2\), x\(_2\)y\(_3\), x\(_3\)y\(_4\) and x\(_4\)y\(_1\) that are shown by the blue arrows in the above image.

(x\(_1\)y\(_2\) + x\(_2\)y\(_3\) + x\(_3\)y\(_4\) + x\(_4\)y\(_1\)) → (1)

Add the diagonal products x\(_2\)y\(_1\), x\(_3\)y\(_2\), x\(_4\)y\(_3\) and x\(_1\)y\(_4\) that are shown by the orange arrows.

(x\(_2\)y\(_1\) + x\(_3\)y\(_2\) + x\(_4\)y\(_3\) + x\(_1\)y\(_4\)) → (2)

Subtract (2) from (1) and multiply the difference by 1/2 to get area of the quadrilateral ABCD.

So, area of the quadrilateral ABCD is given as,

A = (1/2) ⋅ {(x\(_1\)y\(_2\) + x\(_2\)y\(_3\) + x\(_3\)y\(_4\) + x\(_4\)y\(_1\)) - (x\(_2\)y\(_1\) + x\(_3\)y\(_2\) + x\(_4\)y\(_3\) + x\(_1\)y\(_4\))}

Note: We can also calculate the area of a quadrilateral using the coordinates of the vertices by dividing it into two triangles and adding their respective areas. Let us understand this technique using the example given below,

Example: Consider the following four points: A(−3, 1), B(−1, 4), C(3, 2), D(1, −2). These four points are the vertices of a quadrilateral:

Here, we will divide the quadrilateral in two triangles (using either of the diagonals), calculate the (positive value of) the areas of each triangle, and add these values to obtain the total area. In the following figure, quadrilateral ABCD has been divided into ΔABD and ΔADC.

Now, we separately calculate the areas of the two triangles.

Area of Triangle ABC:

= (1/2) |−3 × (4 − 2) + (−1) × (2 − 1) + 3 × (1 − 4)| = (1/2) |−6 −1 −9| = (1/2) × 16 = 8sq.units

Area of Triangle ACD:

= (1/2) |−3 × (−2 − 2) + 1 × (2 − 1) + 3 × (1−(−2))|

= (1/2)|12 + 1 + 9| = (1/2) × 22 = 11sq.units

Area of Quadrilateral ABCD:

Area(ABCD) = Area(ΔABC) + Area(ΔADC) = 8 + 11 = 19 sq.units

What Is the Area of Quadrilateral in Math?

The area of a quadrilateral is the region that is enclosed by it. It is measured in square units such as in², cm², m², etc.

What Is the Area of Quadrilateral Formula?

The area of a quadrilateral can be found by dividing into two triangles using a diagonal. When the diagonal's length and the heights of the two triangles are known, the area (A) of the quadrilateral is, A = (1/2) × Diagonal × (Sum of heights).

What Is the Area of Quadrilateral Formula Using Sides and Angles?

When 4 sides of the quadrilateral a, b, c, and d and the sum of two of its opposite angles θ are known, then its area is found using the formula \(\sqrt{(s-a)(s-b)(s-c)(s-d)-a b c d \cos ^{2} \frac{\theta}{2}}\), where 's' is the semi-perimeter of the quadrilateral. i.e., s = (a + b + c + d)/2.

How To Find the Area of a Quadrilateral Using Heron’s Formula?

We know that the area of a quadrilateral can be found by dividing it into two triangles using a diagonal. Also, we know that the area of a triangle with 3 sides can be found using Heron's formula. Using Heron's formula, the area of the triangle with sides a, b, and c is given by \(\sqrt{s(s-a)(s-b)(s-c)}\), where 's' is the semi-perimeter of the triangle. i.e., s = (a + b + c)/2. Using this formula, we can find the areas of the two triangles (that are formed by the quadrilateral) and add them to get the area of the quadrilateral.

What Are the Formulas To Find the Areas of Different Types of Quadrilaterals?

There are different formulas used to find the areas of different types of quadrilaterals. They are:

• The area of a square of side length 'x' is x².
• The area of a rectangle of dimensions 'l' and 'b' is l × b.
• The area of a parallelogram of base 'b' and height 'h' is b × h.
• The area of a trapezoid whose parallel sides are 'a' and 'b' and height (the perpendicular distance between 'a' and 'b') 'h' is (1/2) (a + b) h.
• The area of a rhombus of diagonals d\(_1\) and d\(_2\) is (1/2) × d\(_1\) × d\(_2\).
• The area of a kite of diagonals d\(_1\) and d\(_2\) is (1/2) × d\(_1\) × d\(_2\).

How To Find the Area of a Quadrilateral With Coordinates?

When a quadrilateral's vertices are given with coordinates, then find the 4 side lengths and the length of a diagonal using the distance formula first. Then divide the quadrilateral into two parts using the diagonal you found, find the area of each triangle using Heron's formula and then add the areas of two triangles which gives the area of the quadrilateral.