How do you find the area of a quadrilateral with four different sides?
The concept of area of a quadrilateral plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. From measuring plots of land to solving questions in geometry or competitive exams, understanding the area of a quadrilateral is a must-have skill for all students.
What Is Area of a Quadrilateral?
A quadrilateral is a four-sided polygon with four vertices and four angles. The area of a quadrilateral refers to the region enclosed by its four sides. This concept is applied in areas such as land measurement, geometry problem-solving, and even coordinate geometry where vertices are given.
Key Formula for Area of Quadrilateral
There isn’t just one formula for the area of a quadrilateral; the formula depends on the type:
| Quadrilateral Type | Area Formula |
|---|---|
| Square | a × a |
| Rectangle | length × breadth |
| Parallelogram | base × height |
| Rhombus/Kite | (1/2) × diagonal₁ × diagonal₂ |
| Trapezium | (1/2) × (sum of parallel sides) × height |
For an irregular quadrilateral (with all sides different), or if you are given four sides and one angle, use Brahmagupta’s formula if the quadrilateral can be inscribed in a circle:
Area = \( \sqrt{(s-a)(s-b)(s-c)(s-d) - abcd \cdot \cos^2(\frac{\theta}{2})} \), where \( s = \frac{a+b+c+d}{2} \) and \( \theta \) is the sum of two opposite angles.
Area from Coordinates (Shoelace Formula)
If the vertices of the quadrilateral are given as coordinates, use this formula:
Area = \( \left|\frac{1}{2} [(x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1) - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1)]\right| \)
Step-by-Step Illustration
- Given: Find the area of a parallelogram with base 10m and height 12m.
Use the formula: Area = base × height
- Substitute: Area = 10 × 12 = 120 m²
Speed Trick or Vedic Shortcut
Here’s a quick tip for rectangles and parallelograms: If the sides are given in centimeters and you want square meters, just multiply and move the decimal four places left.
Example Trick: 200 cm × 300 cm = 60,000 cm² = 6 m²
Tricks like these are covered in Vedantu’s live classes, helping hundreds of students gain confidence and speed.
Solved Example for Irregular Quadrilateral
Find the area of a quadrilateral with sides 5m, 6m, 7m, 8m and one angle of 90° between the first two sides.
1. Find semi-perimeter: s = (5+6+7+8)/2 = 132. Use Brahmagupta’s formula. Here, θ = 90° so cos²(45°) = (1/2).
3. Area = √[(13-5)(13-6)(13-7)(13-8) - 5×6×7×8 × 0.5]
4. Calculate stepwise:
(13-5)=8, (13-6)=7, (13-7)=6, (13-8)=5
Product = 8×7×6×5 = 1680
Subtract: 5×6×7×8×0.5 = 5×6×7×4 = 840
Area = √(1680-840) = √840 ≈ 28.98 m²
Try These Yourself
- Find the area of a rectangle with length 9cm and breadth 7cm.
- If a quadrilateral has vertices (1,2), (6,2), (5,3), (3,4), calculate its area.
- Calculate the area of a trapezium with parallel sides 10m and 20m with height 6m.
- For a rhombus with diagonals 8cm and 10cm, what is the area?
Frequent Errors and Misunderstandings
- Applying the base × height formula to irregular quadrilaterals.
- Forgetting to check if the quadrilateral is cyclic before using Brahmagupta’s formula.
- Mistaking diagonal for side in kite and rhombus area calculations.
- Wrongly listing coordinates in clockwise/counter-clockwise order when using the Shoelace formula (which can reverse the sign).
Relation to Other Concepts
The idea of area of a quadrilateral connects closely with topics such as area of triangle, area of parallelogram, and properties of quadrilaterals. Mastering quadrilateral area helps with learning polygons, mensuration, and coordinate geometry later in your studies.
Classroom Tip
An easy way to remember: “For standard shapes—multiply, for diagonals—halve the product, and for irregular—Brahmagupta saves the day.” Vedantu’s teachers often break area calculations into simple flows and provide visual aids for better memory in live sessions.
We explored area of a quadrilateral—from definition, formula, steps, common mistakes, practice problems, and its links to other important topics. Keep practicing with Vedantu’s resources to become confident in solving area questions quickly and accurately.
Area of Triangle | Quadrilaterals | Area of Parallelogram
FAQs on Area of a Quadrilateral: Formulas, Methods & Examples
1. What's the formula for the area of a quadrilateral?
The area of a quadrilateral depends on the type of quadrilateral and the information provided. For a general (irregular) quadrilateral with known diagonals and the angle between them, you can use the formula:
$$\text{Area} = \frac{1}{2} \times d_1 \times d_2 \times \sin \theta$$
where $d_1$ and $d_2$ are the lengths of the diagonals, and $\theta$ is the angle between them. For special quadrilaterals (like rectangles or parallelograms), there are specific formulas. Vedantu’s resources guide students in choosing the right formula based on the quadrilateral type.
2. How to find area of 4 unequal sides?
To find the area of a quadrilateral with four unequal sides, you need additional information such as the measures of the diagonals or an angle.
- If all four sides and both diagonals are known, use Brahmagupta’s formula for cyclic quadrilaterals:
where $s = \frac{a + b + c + d}{2}$ (the semiperimeter), and $a, b, c, d$ are the lengths of the sides.
- If the quadrilateral is not cyclic, knowing the sides and a diagonal or an angle is necessary. Vedantu’s expert teachers can help clarify which method suits your quadrilateral problem.
3. What is the general formula for a quadrilateral?
The general formula for the area of a quadrilateral when the lengths of the two diagonals and the angle between them are known is:
$$\text{Area} = \frac{1}{2} \times d_1 \times d_2 \times \sin \theta$$
This formula applies to all quadrilaterals as long as $d_1$ and $d_2$ are the lengths of the diagonals, and $\theta$ is the angle between those diagonals. In cases where more specific properties are known (such as for a parallelogram or kite), Vedantu provides detailed lessons on how to apply specialized formulas.
4. What is the area of a 4 side shape?
A 4-sided shape is called a quadrilateral. The area of a quadrilateral depends on its type and measurements. For basic quadrilaterals like rectangles and squares, use:
- Rectangle: $\text{Area} = \text{length} \times \text{breadth}$
- Square: $\text{Area} = (\text{side})^2$
$$\text{Area} = \frac{1}{2} \times d_1 \times d_2 \times \sin \theta$$
Vedantu offers comprehensive learning modules to help students master each formula with solved examples and interactive exercises.
5. How do you calculate the area of a cyclic quadrilateral?
For a cyclic quadrilateral (a quadrilateral inscribed in a circle), the area is calculated using Brahmagupta’s formula:
$$\text{Area} = \sqrt{(s-a)(s-b)(s-c)(s-d)}$$
Here,
- $a, b, c, d$ = sides of the quadrilateral
- $s = \frac{a+b+c+d}{2}$ (semiperimeter)
6. What are the common formulas for the area of special quadrilaterals?
There are special formulas for common quadrilaterals:
- Rectangle: $\text{Area} = \text{length} \times \text{breadth}$
- Square: $\text{Area} = (\text{side})^2$
- Parallelogram: $\text{Area} = \text{base} \times \text{height}$
- Rhombus: $\text{Area} = \frac{1}{2} \times d_1 \times d_2$
- Trapezium: $\text{Area} = \frac{1}{2}(\text{sum of parallel sides}) \times \text{height}$
7. Why is the area calculation different for irregular quadrilaterals?
The area calculation for irregular quadrilaterals varies because their sides and angles are not equal or parallel. This irregularity requires additional information, such as:
- Diagonals and angles
- All side lengths
- The inclusion in a circle (cyclic property)
8. What is the importance of knowing the area of quadrilaterals in real life?
Understanding the area of quadrilaterals has practical applications, including:
- Calculating land plots or fields
- Designing rooms and floors
- Construction and architecture
9. How can Vedantu help me master quadrilateral area problems?
Vedantu offers a wide range of online resources for learning about the area of quadrilaterals, including:
- Live interactive classes with expert teachers
- Downloadable study materials and worksheets
- Doubt-solving sessions
- Practice tests and quizzes
