A 4-sided polygon, or quadrilateral, is a closed shape with exactly four sides, four angles, and four vertices. These polygons are everywhere—from the shape of your computer screen to a piece of land.
4-Sided Polygon Area Calculator
Enter the coordinates of the 4 vertices (in order) to calculate the polygon area using the Shoelace formula.
Coordinates can be entered in any consistent unit; area will be in square of that unit.
Key Characteristics:
- Four sides
- Four interior angles
- The sum of interior angles is always 360°
2. 📐 Properties of Quadrilaterals
| Property | Description |
|---|---|
| Sides | 4 straight, connected sides |
| Angles | 4 interior angles |
| Angle Sum | Always 360° |
| Diagonals | Two diagonals that may or may not be equal |
| Convex/Concave | Can be either, but most are convex |
3. 🧱 Classification of 4-Sided Polygons
Let’s break down the common types of quadrilaterals and their special properties:
| Type | Description | All Angles 90°? | Opposite Sides Equal? |
|---|---|---|---|
| Square | All sides and angles equal | ✅ | ✅ |
| Rectangle | Opposite sides equal; all angles 90° | ✅ | ✅ |
| Rhombus | All sides equal; angles not 90° | ❌ | ✅ |
| Parallelogram | Opposite sides parallel and equal | ❌ | ✅ |
| Trapezoid | Only one pair of parallel sides | ❌ | ❌ |
| Kite | Two pairs of adjacent equal sides | ❌ | ❌ |
| Irregular | No equal sides or angles | ❌ | ❌ |
4. 🧾 General Area Formulas for 4-Sided Polygons
When you don’t know the type, use these general formulas:
A. Bretschneider’s Formula (For any convex quadrilateral):
Area=(s−a)(s−b)(s−c)(s−d)−abcd⋅cos2(A+C2)\text{Area} = \sqrt{(s-a)(s-b)(s-c)(s-d) - abcd \cdot \cos^2\left(\frac{A+C}{2}\right)}Area=(s−a)(s−b)(s−c)(s−d)−abcd⋅cos2(2A+C)
Where:
- a,b,c,da, b, c, da,b,c,d are the side lengths
- A,CA, CA,C are opposite angles
- s=a+b+c+d2s = \frac{a + b + c + d}{2}s=2a+b+c+d
B. Diagonal Method (for cyclic quadrilateral)
Area=12×d1×d2×sin(θ)\text{Area} = \frac{1}{2} \times d_1 \times d_2 \times \sin(\theta)Area=21×d1×d2×sin(θ)
Where:
- d1,d2d_1, d_2d1,d2 = diagonals
- θ\thetaθ = angle between diagonals
5. 📊 Area Formulas by Quadrilateral Type
| Type | Formula | Example |
|---|---|---|
| Square | A=s2A = s^2A=s2 | Side = 5 → A=25A = 25A=25 |
| Rectangle | A=l×wA = l \times wA=l×w | l=8,w=4→A=32l = 8, w = 4 → A = 32l=8,w=4→A=32 |
| Parallelogram | A=b×hA = b \times hA=b×h | b=10,h=3→A=30b = 10, h = 3 → A = 30b=10,h=3→A=30 |
| Rhombus | A=12d1×d2A = \frac{1}{2} d_1 \times d_2A=21d1×d2 | d1=6,d2=4→A=12d_1 = 6, d_2 = 4 → A = 12d1=6,d2=4→A=12 |
| Trapezoid | A=12(a+b)×hA = \frac{1}{2} (a + b) \times hA=21(a+b)×h | a=6,b=4,h=5→A=25a=6, b=4, h=5 → A = 25a=6,b=4,h=5→A=25 |
| Kite | A=12d1×d2A = \frac{1}{2} d_1 \times d_2A=21d1×d2 | d1=5,d2=7→A=17.5d_1 = 5, d_2 = 7 → A = 17.5d1=5,d2=7→A=17.5 |
| Irregular | Use coordinate geometry or divide into triangles | Based on case |
6. 🌍 Real-World Applications
| Field | Application |
|---|---|
| Architecture | Floor plans, roofs, tiles |
| Engineering | Machine parts, bridges |
| Agriculture | Land measurement |
| Interior Design | Carpet/flooring layout |
| Math Education | Geometry, problem solving |
7. 📋 Area Calculation Tables
Table 1: Rectangle Area Examples
| Length (cm) | Width (cm) | Area (cm²) |
|---|---|---|
| 5 | 3 | 15 |
| 8 | 4 | 32 |
| 12 | 6 | 72 |
Table 2: Trapezoid Area Calculation
| Base 1 | Base 2 | Height | Area |
|---|---|---|---|
| 10 | 6 | 4 | 32 |
| 8 | 4 | 5 | 30 |
Table 3: Rhombus Area with Diagonals
| Diagonal 1 | Diagonal 2 | Area |
|---|---|---|
| 6 | 4 | 12 |
| 10 | 5 | 25 |
Table 4: Kite Area
| Diagonal 1 | Diagonal 2 | Area |
|---|---|---|
| 7 | 6 | 21 |
| 5 | 4 | 10 |
Table 5: Square Area
| Side (cm) | Area (cm²) |
|---|---|
| 4 | 16 |
| 9 | 81 |
8. 🖼️ Visuals and Diagrams
A. Shapes Diagram
[Imagine a chart showing all types of quadrilaterals with labels: square, rectangle, trapezoid, etc.]
B. Trapezoid Area Breakdown
A sketch showing the height, base 1, and base 2 with the formula beneath.
C. Coordinate Geometry Example
Show how to split an irregular quadrilateral into two triangles.
9. ❌ Common Mistakes
| Mistake | Correction |
|---|---|
| Using wrong formula for irregular shapes | Use Bretschneider or triangulation |
| Forgetting units | Always include cm², m², etc. |
| Misidentifying the type of quadrilateral | Check sides, angles, and diagonals |
| Assuming diagonals are equal | Only true in squares and rectangles |
| Using side instead of height | Especially in parallelograms and trapezoids |
10. ❓ Frequently Asked Questions
Q1: Can all 4-sided polygons use the same area formula?
No. The formula depends on the type of quadrilateral and known values.
Q2: What if I don’t know the height?
Use diagonals or coordinate geometry, or divide the shape into triangles.
Q3: Are diagonals important for finding area?
Yes, especially for rhombuses, kites, and cyclic quadrilaterals.
Q4: Can irregular quadrilaterals be calculated easily?
You’ll need advanced formulas or a coordinate approach.
Q5: What if my sides are unequal?
Use Bretschneider’s formula or divide the shape into triangles.
11. ✅ Final Thoughts
Understanding how to calculate the area of 4-sided polygons is essential in geometry and real-life applications. Whether you’re working on a school project, designing a house, or measuring farmland, this knowledge will come in handy.
Remember:
- Know your shape type
- Choose the correct formula
- Watch your units
- Double-check values before finalizing calculations
With the right approach, calculating the area of any quadrilateral—regular or irregular—becomes simple and accurate.