Area of General Quadrilateral: Bretschneider Formula, Heron’s Formula, Examples

The area of a quadrilateral can be defined as a surface enclosed within a set of four sides. A quadrilateral is a closed polygon that could be regular or irregular depending on whether the sides are equal or not. A quadrilateral is made up of four sides, four vertices, and four angles. Quadrilaterals can be classified into six types based on their properties. The six types of quadrilaterals are as follows: 

• Rectangle
• Square
• Parallelogram
• Rhombus
• Trapezium’
• Kite

There are different methods to calculate the area of a quadrilateral based on the information provided. This article explains in detail the different methods and formulae for measuring the area of the quadrilateral. The article also touches upon the properties of the different quadrilaterals and the formulas to find their areas. Let us discuss the general quadrilateral, its properties along with the different methods to calculate its area.

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What is a Quadrilateral?

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A Quadrilateral can be defined as a closed polygon enclosed by four equal or unequal sides. In short, a quadrilateral is a four-sided closed polygon.

Some of the properties of a quadrilateral are as follows:

• A quadrilateral must have four sides, four vertices, and four angles.
• The sum of the four angles of a quadrilateral must be equal to 360^o.
• There can be only two diagonals in a quadrilateral.

A quadrilateral may be classified as regular or irregular depending on the sides of the polygon. If all the four sides of the quadrilateral are equal, then it is said to be a regular quadrilateral. If the sides of the quadrilateral are not equal, then it is said to be an irregular quadrilateral.

Quadrilaterals are broadly classified into six types based on their properties. The six types of quadrilaterals are as follows:

1. Rectangle
2. Square
3. Parallelogram
4. Rhombus
5. Trapezium
6. Kite

Since square and rhombus have sides that are equal in length, they are called regular quadrilaterals while the rest are irregular quadrilaterals.

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Area of a Quadrilateral

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The area of a quadrilateral can be defined as the amount of space enclosed within the sides of a quadrilateral. Area is calculated for two-dimensional shapes and it is measured in square units. Any value depicting the area of a surface is denoted using the units like m², cm², sq. km., sq. ft., etc.

Several methods are used to measure the area of a quadrilateral. The area of a quadrilateral like square, rectangle, or parallelogram can be calculated by directly using the formulas. However, the method or formula to be applied for finding the area of a quadrilateral must be determined based on the information provided and the type of the quadrilateral.

For finding the area of a quadrilateral like square, rectangle, parallelogram, and rhombus, it is always advisable to apply the respective formulas directly. However, for finding the area of a general quadrilateral, there are different methods that can be followed. The area of a general quadrilateral can be determined using Heron’s formula or Bretschneider’s formula or splitting the quadrilateral into two triangles or by using coordinate geometry.

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Area of a Quadrilateral by Splitting it into Two Triangles

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The area of a quadrilateral can be measured by splitting it into two triangles. If the length of the diagonal and the heights of the two triangles are known then it is the best method to use for finding the area of the quadrilateral. In this method, the area of the two triangles are calculated separately and then the two areas are summed up to obtain the area of the quadrilateral.

Let us consider the following example to get a clear idea. Consider the quadrilateral ABCD.

In the

\(\Box\)ABCD, let the diagonal of the quadrilateral be denoted by ‘d’. The quadrilateral ABCD is divided into two triangles namely ΔADC and ΔABC. The heights of the two triangles are h1 and h2 respectively. The area of the quadrilateral can be determined by adding the areas of the two triangles ΔADC and ΔABC.

⇒ Area of

\(\Box\)ABCD = Area of ΔADC + Area of ΔABC

⇒ Area of

\(\Box\)ABCD = [( ½) × length of diagonal × height of triangle] + [(½) × length of diagonal × height of triangle]

⇒ Area of

\(\Box\)ABCD = [(½) × d ×h1] + [(½) × d × h2]

⇒ Area of

\(\Box\)ABCD = (½) × d × (h1 + h2)

Therefore we can infer that area of a quadrilateral can be obtained by using the below formula

 (½) × Length of the diagonal × (sum of the heights of the triangles) provided the length of a diagonal and heights are given.

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Area of a Quadrilateral using Heron’s Formula

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Area of a quadrilateral can be determined by splitting the quadrilateral into two triangles when the length of the diagonal and heights of the triangles are provided. However, in some situations, these details are not available. Sometimes, the lengths of the four sides of the quadrilateral and the length of the diagonal are given. In such situations, the Heron’s formula should be used.

Consider a quadrilateral ABCD with a diagonal AC. The diagonal AC divides the quadrilateral into two triangles.

Here the length of the diagonal AC and the lengths of the sides are given. Let AB = a, BC = b, CD = c, DA = d and AC = e.

Steps to find the area of the quadrilateral with the above information:

1. Find the semi-perimeter of the ΔABC and ΔADC.

s1 = (AB + BC + CA)/2

⇒ s1 = (a + b + e)/2

Similarly for

ΔADC, s2 = (AD + DC+ CA)/2

⇒ s2 = (d + c + e)/2

1. The area of the quadrilateral ABCD = area of ΔABC + area of ΔADC

⇒ Area of

?ABCD = [√s1(s1 - a)(s1 - b)(s1 - e)] + [√s2(s2 - d)(s2 - c)(s2 - e)]

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Area of a Quadrilateral using Bretschneider’s Formula

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Area of a quadrilateral can be determined when the lengths of the sides and the opposite angles are given using the Bretschneider’s formula. Consider the quadrilateral ABCD with lengths of sides as p,q,r, and s. The opposite angles ∠A and ∠C are denoted as

θ₁ and θ₂ respectively.

Steps to find the area of a quadrilateral using the Bretschneider’s formula are as follows:

1. Find the sum of the opposite angles. 

⇒

 θ = θ₁ + θ₂

1. Find the semi-perimeter of the quadrilateral.

⇒ s₁= (p + q + r + s)/2

1. Now apply the Bretschneider’s formula as follows:

√(s₁ - p) (s₁ - q) (s₁ - r) (s₁-s) -  pqrs cos²(θ/2).

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Area of a Quadrilateral using Coordinate Geometry 

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The area of a quadrilateral can be determined when the coordinates of vertices of the quadrilateral are given. Consider a quadrilateral ABCD where the coordinates of the vertices are represented as

A(x₁,y₁), B(x₂,y₂), C(x₃,y₃), and D(x₄,y₄).

Steps to find the area of quadrilateral using coordinate geometry principles:

1.The coordinates of the quadrilateral should be written as follows

Quadr

2.Add the products of the coordinates indicated by the red line as follows:

⇒ x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁

3.Now, add the products of the coordinates indicated by the blue line as follows:

⇒x₂y₁ + x₃y₂ + x₄y₃ + x₁y₄

4.Subtract the sum obtained in Step 3 from the sum obtained in Step 2 and divide the result by 2. This step can be represented as follows:

(½)[(x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) - (x₂y₁ + x₃y₂ + x₄y₃ + x₁y₄)]

Thus the formula to find the area of a quadrilateral using this method can be represented as(½)[(x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) - (x₂y₁ + x₃y₂ + x₄y₃ + x₁y₄)].

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Area of Different Types of Quadrilaterals

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The areas of some of the quadrilaterals can be calculated using formulas. Mentioned in the table below are the formulas for calculating areas of some quadrilaterals.

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Solved Problems

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• In a \(\Box\)ABCD, the length of the diagonal BD is 20 cm and the heights of the ΔADC and ΔABC are 6 cm and 8 cm respectively. Find the area of the quadrilateral ABCD.

Solution:

Given:

Length of diagonal b = 20 cm

Height of

ΔADC, h1 = 6 cm

Height of

ΔABC, h2 =8 cm

Area of \(\Box\)ABCD = (½) × b × (h1 + h2)

⇒ Area of \(\Box\)ABCD = (½) × 20 × (6 + 8)

= (½) × 20 × 14

= 140 sq. cm.

Area of \(\Box\)ABCD = 140 sq.cm.

• Find the area of a quadrilateral with vertices (7, 5), (4,10), (-6,11) and (-5,2).

Solution:

Let A(7, 5), B(4,10), C(-6,11) and D(-5,2) be the vertices of the quadrilateral ABCD.

Let A(7, 5) = A(x₁, y₁)

Let B(4,10) = B(x₂,y₂)

Let C(-6,11) = C(x₃,y₃)

Let D(-5,2) = D(x₄,y₄)

Area of \(\Box\)ABCD = (½)[(x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) - (x₂y₁ + x₃y₂ + x₄y₃ + x₁y₄)]

= (½){ [(7 × 10) + (4 × 11) + (-6 × 2) + (-5 × 5)] - [(5 × 4) + (-6 × 10) + ( -5 × 11) + (7 × 2)]}

= (½)[(70 + 44 - 12 - 25) - (20 - 60 - 55 +14)]

=(½)[77 - (-109)]

=(½)(186) 

= 93 sq.cm.

Thus area of \(\Box\)ABCD = 93 sq.cm.

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Things to Remember

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• A quadrilateral is a four-sided closed figure.
• Area of a quadrilateral can be obtained through different ways depending on the information provided. Area can be calculated by dividing the quadrilateral into two triangles or through Heron’s formula or through Bretschneider’s formula or through principles of cor-ordinate geometry.
• The area of some of the quadrilaterals like rectangle, square, parallelogram, rhombus, trapezium, and kite can be calculated using the formulas.
• Heron’s formula is used when the sides of a quadrilateral are given along with the diagonal length.
• Bretschneider’s formula can be used when the four sides of the quadrilateral are given along with the opposite angles.