Quadrilateral Area Calculator
Calculated Area:
'; resultDiv.innerHTML += 'The area of the quadrilateral is: ' + area.toFixed(4) + ' square units'; }Understanding the Area of a Quadrilateral
A quadrilateral is a polygon with four sides and four vertices (corners). Quadrilaterals are fundamental shapes in geometry, encompassing a wide variety of forms such as squares, rectangles, parallelograms, rhombuses, trapezoids, and kites, as well as irregular quadrilaterals that don't fit into these specific categories.
The General Formula for Quadrilateral Area
While specific formulas exist for special types of quadrilaterals (e.g., length × width for a rectangle, base × height for a parallelogram), a powerful general formula can calculate the area of any convex quadrilateral if you know the lengths of its two diagonals and the angle between them. This formula is particularly useful for irregular quadrilaterals where side lengths alone might not be sufficient.
The formula is:
Area = 0.5 × d1 × d2 × sin(θ)
Where:
d1is the length of the first diagonal.d2is the length of the second diagonal.θ(theta) is the angle (in degrees or radians) between the two diagonals.sin(θ)is the sine of that angle.
This formula works because any convex quadrilateral can be divided into two triangles by each diagonal. The area calculation essentially sums the areas of these two triangles, using the property that the area of a triangle is 0.5 × a × b × sin(C), where 'a' and 'b' are two sides and 'C' is the included angle.
How to Use the Calculator
Our Quadrilateral Area Calculator simplifies this process for you:
- Length of Diagonal 1: Enter the length of the first diagonal of your quadrilateral. This can be any positive numerical value.
- Length of Diagonal 2: Enter the length of the second diagonal. This can also be any positive numerical value.
- Angle Between Diagonals (in degrees): Input the angle formed at the intersection of the two diagonals. This angle must be between 0 and 180 degrees. If the diagonals are perpendicular (as in a square, rhombus, or kite), the angle is 90 degrees.
- Click "Calculate Area" to see the result in square units.
Examples
Example 1: A Rhombus
A rhombus has diagonals that are perpendicular (angle = 90 degrees). Let's say a rhombus has diagonals of 12 units and 10 units.
- Diagonal 1 (d1) = 12
- Diagonal 2 (d2) = 10
- Angle (θ) = 90 degrees
Using the formula: Area = 0.5 × 12 × 10 × sin(90°)
Since sin(90°) = 1,
Area = 0.5 × 12 × 10 × 1 = 60 square units
Example 2: A General Quadrilateral
Consider an irregular quadrilateral where the diagonals are 15 units and 8 units, and they intersect at an angle of 60 degrees.
- Diagonal 1 (d1) = 15
- Diagonal 2 (d2) = 8
- Angle (θ) = 60 degrees
Using the formula: Area = 0.5 × 15 × 8 × sin(60°)
Since sin(60°) ≈ 0.8660,
Area = 0.5 × 15 × 8 × 0.8660 = 60 × 0.8660 ≈ 51.96 square units
Example 3: A Square
A square with side length 's' has diagonals of length s√2, and they are perpendicular. If a square has a side length of 7 units, its diagonals are 7√2 ≈ 9.899 units each.
- Diagonal 1 (d1) = 9.899
- Diagonal 2 (d2) = 9.899
- Angle (θ) = 90 degrees
Using the formula: Area = 0.5 × 9.899 × 9.899 × sin(90°)
Area = 0.5 × 97.990201 × 1 ≈ 48.995 square units
Note: The actual area of a square with side 7 is 7 × 7 = 49. The slight difference is due to rounding the diagonal length. If we use s√2 directly:
Area = 0.5 × (s√2) × (s√2) × sin(90°) = 0.5 × s² × 2 × 1 = s². This confirms the formula's accuracy for a square.