Find The Area Of A Quadrilateral Calculator

Calculate Area by Coordinates (Shoelace Formula)

Enter the coordinates of the four vertices (x1, y1), (x2, y2), (x3, y3), (x4, y4) of the quadrilateral in order (clockwise or counter-clockwise).

Vertex	X	Y	x_i * y_{i+1}	y_i * x_{i+1}
V1 (x1,y1)	1	1	0	0
V2 (x2,y2)	6	2	0	0
V3 (x3,y3)	5	7	0	0
V4 (x4,y4)	2	5	0	0

Table showing vertex coordinates and terms for Shoelace formula.

What is an Area of a Quadrilateral Calculator?

An area of a quadrilateral calculator is a tool used to determine the surface area enclosed by a four-sided polygon (a quadrilateral). Quadrilaterals can be simple (like squares, rectangles, parallelograms, trapezoids, rhombuses, and kites) or complex and irregular. This calculator specifically uses the coordinates of the four vertices to find the area using the Shoelace formula, making it suitable for any simple quadrilateral, including irregular ones.

Anyone needing to find the area of a four-sided plot of land, a room, a material sheet, or any other four-sided shape can use an area of a quadrilateral calculator. This includes students, engineers, architects, land surveyors, and DIY enthusiasts.

A common misconception is that you always need side lengths and angles. While some formulas use those, knowing the coordinates of the vertices is often a more direct way to find the area of any simple quadrilateral, especially irregular ones, using the area of a quadrilateral calculator with the Shoelace method.

Area of a Quadrilateral Calculator Formula and Mathematical Explanation

There are several formulas to calculate the area of a quadrilateral, depending on what information is known:

• For Special Quadrilaterals:
  • Rectangle: Area = length × width
  • Square: Area = side × side
  • Parallelogram: Area = base × height
  • Trapezoid: Area = 0.5 × (base1 + base2) × height
  • Rhombus/Kite: Area = 0.5 × diagonal1 × diagonal2
• Using Diagonals and the Angle Between Them: Area = 0.5 × d1 × d2 × sin(θ), where d1 and d2 are the lengths of the diagonals and θ is the angle between them.
• Bretschneider's Formula (using four sides and two opposite angles): For a general quadrilateral with sides a, b, c, d, and semi-perimeter s = (a+b+c+d)/2, and opposite angles α and γ, Area = √((s-a)(s-b)(s-c)(s-d) – abcd × cos²((α+γ)/2)).
• Shoelace Formula (using coordinates of vertices): This is the method our area of a quadrilateral calculator uses. If the vertices are (x1, y1), (x2, y2), (x3, y3), and (x4, y4) in order (clockwise or counter-clockwise), the area is:
  Area = 0.5 * |(x1y2 + x2y3 + x3y4 + x4y1) – (y1x2 + y2x3 + y3x4 + y4x1)|
  Let Sum1 = (x1y2 + x2y3 + x3y4 + x4y1) and Sum2 = (y1x2 + y2x3 + y3x4 + y4x1).
  Then Area = 0.5 * |Sum1 – Sum2|.

Variables Table (Shoelace Formula)

Variable	Meaning	Unit	Typical Range
(x1, y1)	Coordinates of the first vertex	Length units (e.g., m, cm, ft)	Any real number
(x2, y2)	Coordinates of the second vertex	Length units	Any real number
(x3, y3)	Coordinates of the third vertex	Length units	Any real number
(x4, y4)	Coordinates of the fourth vertex	Length units	Any real number
Area	Area enclosed by the quadrilateral	Square length units (e.g., m², cm², ft²)	Non-negative real number

Variables used in the Shoelace formula for the area of a quadrilateral.

Practical Examples (Real-World Use Cases)

Example 1: Irregular Plot of Land

An architect is surveying an irregular plot of land defined by four corner points with the following coordinates (in meters): V1=(0,0), V2=(50,10), V3=(40,60), V4=(5,50).

Using the Shoelace formula with our area of a quadrilateral calculator:

• x1=0, y1=0
• x2=50, y2=10
• x3=40, y3=60
• x4=5, y4=50

Sum1 = (0*10 + 50*60 + 40*50 + 5*0) = 0 + 3000 + 2000 + 0 = 5000

Sum2 = (0*50 + 10*40 + 60*5 + 50*0) = 0 + 400 + 300 + 0 = 700

Area = 0.5 * |5000 – 700| = 0.5 * 4300 = 2150 square meters.

Example 2: Cutting a Piece of Material

A designer needs to cut a quadrilateral piece of fabric from a larger sheet. The corners are marked at coordinates (inches): A=(2,3), B=(10,5), C=(8,12), D=(3,9).

Using the area of a quadrilateral calculator:

• x1=2, y1=3
• x2=10, y2=5
• x3=8, y3=12
• x4=3, y4=9

Sum1 = (2*5 + 10*12 + 8*9 + 3*3) = 10 + 120 + 72 + 9 = 211

Sum2 = (3*10 + 5*8 + 12*3 + 9*2) = 30 + 40 + 36 + 18 = 124

Area = 0.5 * |211 – 124| = 0.5 * 87 = 43.5 square inches.

How to Use This Area of a Quadrilateral Calculator

1. Enter Coordinates: Input the X and Y coordinates for each of the four vertices (V1, V2, V3, V4) of the quadrilateral into the respective fields (x1, y1, x2, y2, x3, y3, x4, y4). Ensure the vertices are entered in either clockwise or counter-clockwise order.
2. Calculate: The calculator will automatically update the area and intermediate sums (Sum1, Sum2) as you type. You can also click the "Calculate Area" button.
3. View Results: The primary result is the calculated "Area" in square units. You also see "Sum1", "Sum2", and their absolute difference, which are intermediate steps in the Shoelace formula.
4. Visualize: The SVG chart below the results visually represents the quadrilateral based on your input coordinates (scaled to fit).
5. Table Data: The table shows the input coordinates and the individual terms contributing to Sum1 and Sum2.
6. Reset: Click "Reset" to clear the fields and return to default values.
7. Copy: Click "Copy Results" to copy the main area, intermediate values, and input coordinates to your clipboard.

Understanding the area is crucial for tasks like estimating materials, land valuation, or design specifications. An accurate area of a quadrilateral calculator provides this quickly.

Key Factors That Affect Area of a Quadrilateral Results

1. Vertex Coordinates: The most direct factor. Changing the x or y value of any vertex will change the shape and thus the area.
2. Order of Vertices: While the Shoelace formula works with either clockwise or counter-clockwise order, mixing the order or inputting vertices randomly will result in an incorrect area or the area of a self-intersecting polygon.
3. Side Lengths: Although not direct inputs for this calculator, the side lengths are determined by the coordinates, and thus affect the area.
4. Interior Angles: Similarly, the angles are determined by the vertex positions and significantly influence the area. A more "spread out" quadrilateral will generally have a larger area than a "squashed" one with the same perimeter.
5. Whether the Quadrilateral is Convex or Concave: The Shoelace formula works for both simple convex and concave quadrilaterals, as long as the vertices are listed sequentially around the perimeter.
6. Units Used: The area will be in square units of the units used for the coordinates (e.g., if coordinates are in meters, the area is in square meters). Consistency is key.

Frequently Asked Questions (FAQ)

1. What is the Shoelace formula for the area of a quadrilateral?

The Shoelace formula calculates the area of a simple polygon (including a quadrilateral) given the coordinates of its vertices. For a quadrilateral with vertices (x1, y1), (x2, y2), (x3, y3), (x4, y4) listed sequentially, Area = 0.5 * |(x1y2 + x2y3 + x3y4 + x4y1) – (y1x2 + y2x3 + y3x4 + y4x1)|. Our area of a quadrilateral calculator uses this.

2. Does the order of vertices matter in this area of a quadrilateral calculator?

Yes, the vertices must be entered in sequential order as you move around the perimeter, either clockwise or counter-clockwise. A random order will likely give an incorrect area for a simple quadrilateral.

3. Can this calculator find the area of an irregular quadrilateral?

Yes, the Shoelace formula is particularly useful for irregular quadrilaterals, as long as you know the coordinates of the vertices.

4. What if my quadrilateral is a square or rectangle?

The calculator will still work perfectly. If you input the coordinates of a square or rectangle, it will give you the correct area (length × width or side²).

5. Can I use this calculator for a self-intersecting (crossed) quadrilateral?

The Shoelace formula, as applied here for a simple polygon, assumes the quadrilateral does not self-intersect. For self-intersecting quadrilaterals, the interpretation of "area" can be different, and the formula might give the sum/difference of areas of the enclosed triangles.

6. What units should I use for the coordinates?

You can use any unit of length (meters, feet, inches, cm, etc.), but be consistent. The area will be in the square of that unit (m², ft², in², cm²).

7. What if I only know the side lengths and angles, not coordinates?

If you know all four side lengths and two opposite angles, you could use Bretschneider's formula. If it's a special quadrilateral like a parallelogram or trapezoid, you can use formulas involving base and height or diagonals. This specific area of a quadrilateral calculator requires coordinates.

8. How accurate is this area of a quadrilateral calculator?

The calculation is as accurate as the input coordinates you provide. The mathematical formula is exact for a simple quadrilateral defined by those vertices.

Related Tools and Internal Resources

• Area of a Triangle Calculator: Calculate the area of a triangle using various methods.
• Rectangle Area Calculator: Specifically for rectangles, using length and width.
• Parallelogram Area Calculator: Find the area of a parallelogram using base and height or sides and angle.
• Trapezoid Area Calculator: Calculate the area of a trapezoid given its bases and height.
• Polygon Area Calculator: A more general tool for polygons with more than four sides using coordinates (Shoelace formula).
• Coordinate Geometry Tools: Other tools related to coordinate geometry, like distance and midpoint calculators.