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 is the area of a quadrilateral?
The area of a quadrilateral is the amount of space enclosed within its four sides. Different formulas are used to calculate the area, depending on the type of quadrilateral (e.g., square, rectangle, parallelogram, trapezium, rhombus, kite, irregular quadrilateral).
2. How do I find the area of a quadrilateral if all four sides are different?
For a quadrilateral with four unequal sides, you'll typically use Brahmagupta's formula. This formula requires knowing the lengths of all four sides (a, b, c, d) and one diagonal (e) or the angle between two sides. The formula is complex, so using a calculator is often recommended.
3. What is the formula for the area of a cyclic quadrilateral?
For a cyclic quadrilateral (a quadrilateral whose vertices all lie on a single circle), Brahmagupta's formula can be used. Alternatively, if you know the lengths of the sides (a, b, c, d) and the length of one diagonal, you can use alternative methods by dividing it into two triangles.
4. What is Brahmagupta's formula?
Brahmagupta's formula calculates the area (A) of a cyclic quadrilateral given the lengths of its four sides (a, b, c, d):
A = √[(s-a)(s-b)(s-c)(s-d)], where s is the semi-perimeter, calculated as s = (a+b+c+d)/2. This formula is an extension of Heron's formula for triangles.
5. How do I calculate the area of a quadrilateral using its coordinates?
The area of a quadrilateral with vertices at coordinates (x₁, y₁), (x₂, y₂), (x₃, y₃), and (x₄, y₄) can be found using the shoelace theorem (also known as Gauss's area formula). The formula is: A = ½| (x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) - (y₁x₂ + y₂x₃ + y₃x₄ + y₄x₁) |
6. What are the area formulas for different types of quadrilaterals?
The area formulas vary depending on the quadrilateral type:
• **Square:** side²
• **Rectangle:** length × width
• **Parallelogram:** base × height
• **Rhombus:** ½ × diagonal₁ × diagonal₂
• **Trapezium:** ½ × (sum of parallel sides) × height
• **Kite:** ½ × diagonal₁ × diagonal₂
7. How can I find the area of an irregular quadrilateral?
For irregular quadrilaterals, you can use methods such as dividing the quadrilateral into two triangles using a diagonal and applying Heron's formula to each triangle; or using coordinates and the shoelace theorem.
8. Can I use Heron's formula to find the area of any quadrilateral?
No, Heron's formula is specifically for triangles. However, you can adapt it by dividing a quadrilateral into two triangles. You'll need to know the lengths of the three sides of each triangle.
9. What are some common mistakes students make when calculating quadrilateral areas?
Common errors include using the wrong formula for the type of quadrilateral, incorrect measurement of sides or angles, calculation mistakes, and forgetting units. Carefully identify the type of quadrilateral and use the appropriate formula. Double-check your calculations and remember to state the units (e.g., cm², m²).
10. How is the area of a quadrilateral used in real-world applications?
Calculating quadrilateral areas is crucial in various fields:
• **Land surveying:** Determining property sizes
• **Construction:** Planning building layouts and material requirements
• **Architecture:** Designing floor plans and spaces
• **Engineering:** Calculating volumes and surface areas
11. What's the difference between a cyclic and non-cyclic quadrilateral?
A cyclic quadrilateral has all its vertices lying on a single circle. A non-cyclic quadrilateral does not meet this condition. Different formulas are used to calculate the area depending on the type.
12. How does the order of coordinates affect the area calculation using the shoelace theorem?
When using the shoelace theorem, the order of the coordinates matters. Listing coordinates clockwise will yield a negative area, while counter-clockwise order gives a positive area. Always use consistent order and take the absolute value of the result to obtain the area.
