Quadrangle Area Calculator

Quadrangle Area Calculator

Enter the lengths of the four sides and one diagonal of the quadrangle to calculate its area. The diagonal must connect the vertex between Side 1 and Side 2 to the vertex between Side 3 and Side 4, effectively dividing the quadrangle into two triangles.











Result:

function calculateQuadrangleArea() { var side1 = parseFloat(document.getElementById('side1Length').value); var side2 = parseFloat(document.getElementById('side2Length').value); var side3 = parseFloat(document.getElementById('side3Length').value); var side4 = parseFloat(document.getElementById('side4Length').value); var diagonal = parseFloat(document.getElementById('diagonalLength').value); var resultDiv = document.getElementById('quadrangleAreaResult'); resultDiv.innerHTML = "; // Clear previous results // Input validation if (isNaN(side1) || isNaN(side2) || isNaN(side3) || isNaN(side4) || isNaN(diagonal) || side1 <= 0 || side2 <= 0 || side3 <= 0 || side4 <= 0 || diagonal <= 0) { resultDiv.innerHTML = 'Please enter valid positive numbers for all lengths.'; return; } // Heron's formula for triangle area function calculateTriangleArea(a, b, c) { // Check triangle inequality theorem if (a + b <= c || a + c <= b || b + c <= a) { return NaN; // Not a valid triangle } var s = (a + b + c) / 2; var area = Math.sqrt(s * (s – a) * (s – b) * (s – c)); return area; } // Calculate area of the first triangle (sides: side1, side2, diagonal) var area1 = calculateTriangleArea(side1, side2, diagonal); // Calculate area of the second triangle (sides: side3, side4, diagonal) var area2 = calculateTriangleArea(side3, side4, diagonal); if (isNaN(area1) || isNaN(area2)) { resultDiv.innerHTML = 'The given side and diagonal lengths do not form a valid quadrangle (triangle inequality violated for one or both internal triangles).'; return; } var totalArea = area1 + area2; resultDiv.innerHTML = '

Total Quadrangle Area: ' + totalArea.toFixed(4) + ' square units

'; }

Understanding Quadrangles and Their Area

What is a Quadrangle?

A quadrangle, also known as a quadrilateral, is a polygon with four sides (edges) and four vertices (corners). It is one of the most fundamental shapes in geometry, found everywhere from architectural designs to natural patterns. The sum of the interior angles of any quadrangle is always 360 degrees.

Types of Quadrangles

Quadrangles can be classified into several types based on their side lengths, angle measures, and parallelism of sides:

  • Square: All four sides are equal in length, and all four angles are right angles (90 degrees).
  • Rectangle: Opposite sides are equal and parallel, and all four angles are right angles.
  • Rhombus: All four sides are equal in length, and opposite angles are equal. Diagonals bisect each other at right angles.
  • Parallelogram: Opposite sides are equal and parallel, and opposite angles are equal.
  • Trapezoid (or Trapezium): Has at least one pair of parallel sides.
  • Isosceles Trapezoid: A trapezoid where the non-parallel sides are equal in length, and base angles are equal.
  • Kite: Has two pairs of equal-length sides that are adjacent to each other. Its diagonals are perpendicular.
  • Irregular Quadrangle: A quadrangle that does not fit into any of the above specific categories. Its sides and angles can all be different.

Calculating the Area of a General Quadrangle

While specific formulas exist for the area of squares, rectangles, parallelograms, and trapezoids, calculating the area of a general or irregular quadrangle requires a more versatile approach. One of the most common and practical methods is to divide the quadrangle into two triangles using one of its diagonals.

Consider a quadrangle with sides of lengths s1, s2, s3, and s4. If we draw a diagonal of length d that connects the vertex between s1 and s2 to the vertex between s3 and s4, we effectively create two triangles:

  1. Triangle 1: Formed by sides s1, s2, and the diagonal d.
  2. Triangle 2: Formed by sides s3, s4, and the diagonal d.

The total area of the quadrangle is simply the sum of the areas of these two triangles.

Heron's Formula for Triangle Area

To find the area of each triangle, we can use Heron's formula, which is particularly useful when only the side lengths of a triangle are known. For a triangle with sides a, b, and c:

  1. First, calculate the semi-perimeter (s) of the triangle: s = (a + b + c) / 2
  2. Then, apply Heron's formula to find the area (A): A = √(s * (s - a) * (s - b) * (s - c))

This formula works for any triangle, provided the side lengths form a valid triangle (i.e., the sum of any two sides must be greater than the third side).

Example Calculation

Let's calculate the area of a quadrangle with the following dimensions:

  • Side 1 (s1): 5 units
  • Side 2 (s2): 6 units
  • Side 3 (s3): 7 units
  • Side 4 (s4): 8 units
  • Diagonal (d): 9 units

Step 1: Calculate Area of Triangle 1 (sides 5, 6, 9)

  • Semi-perimeter (s1_p) = (5 + 6 + 9) / 2 = 20 / 2 = 10
  • Area 1 = √(10 * (10 – 5) * (10 – 6) * (10 – 9))
  • Area 1 = √(10 * 5 * 4 * 1) = √200 ≈ 14.1421 square units

Step 2: Calculate Area of Triangle 2 (sides 7, 8, 9)

  • Semi-perimeter (s2_p) = (7 + 8 + 9) / 2 = 24 / 2 = 12
  • Area 2 = √(12 * (12 – 7) * (12 – 8) * (12 – 9))
  • Area 2 = √(12 * 5 * 4 * 3) = √720 ≈ 26.8328 square units

Step 3: Sum the Areas

  • Total Quadrangle Area = Area 1 + Area 2
  • Total Quadrangle Area = 14.1421 + 26.8328 ≈ 40.9749 square units

This method provides a reliable way to determine the area of any general quadrangle, given its four side lengths and the length of one of its diagonals.

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