1. Understanding Area of a Quadrilateral
A quadrilateral is any four-sided closed figure, and its area depends on its shape and the information given. Unlike rectangles or squares, a general quadrilateral does not have a single formula. Instead, we choose the method based on sides, height, or diagonals.
Most area calculations involve breaking the quadrilateral into two triangles or using special formulas when the quadrilateral has symmetry or parallel sides.
2. Area of Basic Quadrilaterals
Special quadrilaterals have simple and direct formulas for area. These are the ones used most often in school geometry and mensuration.
2.1. Rectangle
If length = \(l\) and breadth = \(b\), then:
\(\text{Area} = l \times b\)
2.2. Square
If each side = \(a\), then:
\(\text{Area} = a^2\)
2.3. Parallelogram
If base = \(b\) and height = \(h\), then:
\(\text{Area} = b \times h\)
2.4. Rhombus
If diagonals are \(d_1\) and \(d_2\), then:
\(\text{Area} = \dfrac{1}{2} d_1 d_2\)
2.5. Trapezium
If parallel sides (bases) are \(a\) and \(b\), and height = \(h\):
\(\text{Area} = \dfrac{1}{2}(a + b)h\)
2.6. Kite
If diagonals are \(d_1\) and \(d_2\):
\(\text{Area} = \dfrac{1}{2} d_1 d_2\)
3. General Strategy: Splitting into Two Triangles
For a general quadrilateral with no special properties, the easiest method is to divide it into two triangles by drawing a diagonal.
For quadrilateral \(ABCD\), draw diagonal \(AC\). Then:
\(\text{Area of } ABCD = \text{Area}(\triangle ABC) + \text{Area}(\triangle ACD)\)
Each triangle’s area can be found using:
- \(\dfrac{1}{2} \times \text{base} \times \text{height}\)
- Heron's formula (if all sides known)
- Right-triangle method (if one triangle is right-angled)
4. Using Diagonals to Find Area
Some quadrilaterals have special diagonal properties that help calculate area directly.
4.1. When Diagonals are Perpendicular
If diagonals \(d_1\) and \(d_2\) intersect at right angles (as in a rhombus or kite):
\(\text{Area} = \dfrac{1}{2} d_1 d_2\)
4.2. When Quadrilateral is Cyclic or Special
Cyclic quadrilaterals and some coordinate geometry quadrilaterals allow alternative area formulas, but these are usually introduced in higher classes.
5. Choosing the Right Formula
Different problems require different approaches. Here is how you decide quickly.
5.1. If Parallel Sides are Given
Use trapezium formula: \(\dfrac{1}{2}(a + b)h\).
5.2. If Diagonals are Known
Check if diagonals are perpendicular or if the quadrilateral is a rhombus/kite. Use:
\(\dfrac{1}{2} d_1 d_2\)
5.3. If Nothing Special is Given
Always split the quadrilateral into two triangles and calculate their areas separately.
6. Examples Inside Concepts
Here are a few simple examples to help connect the ideas.
6.1. Example 1: Using Base and Height
A parallelogram with base \(12\text{ cm}\) and height \(7\text{ cm}\):
\(\text{Area} = 12 \times 7 = 84\text{ cm}^2\)
6.2. Example 2: Using Diagonals
A rhombus with diagonals \(10\text{ cm}\) and \(24\text{ cm}\):
\(\text{Area} = \dfrac{1}{2}(10)(24) = 120\text{ cm}^2\)
6.3. Example 3: Splitting into Triangles
Quadrilateral \(ABCD\) is split into triangles by diagonal \(AC\). If:
- Area of \(\triangle ABC = 30\text{ cm}^2\)
- Area of \(\triangle ACD = 25\text{ cm}^2\)
Then:
\(\text{Total Area} = 30 + 25 = 55\text{ cm}^2\)