Enter the coordinates (x, y) for each vertex of your polygon in order.
Area: —
Understanding Polygon Area Calculation
A polygon is a closed shape made of straight line segments. Calculating the area of a polygon is a fundamental concept in geometry with applications in fields like surveying, computer graphics, engineering, and architecture. For a polygon with more than three sides, simply knowing the lengths of the sides is not enough to determine its area; the order and angles of the vertices are crucial.
The Shoelace Formula
The most common and efficient method for calculating the area of any simple polygon (one that does not intersect itself) given its vertices is the Shoelace Formula, also known as the surveyor's formula or Gauss's area formula.
Given the coordinates of the vertices of a polygon in order (either clockwise or counterclockwise) as $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$, the area $A$ is calculated as:
This formula can be visualized by listing the coordinates vertically and drawing diagonal "shoelaces". You multiply diagonally downwards and to the right, summing these products. Then, you multiply diagonally upwards and to the right, summing those products. The absolute difference between these two sums, divided by 2, gives the area.
How the Calculator Works:
This calculator implements the Shoelace Formula. You provide the number of vertices for your polygon and then input the (x, y) coordinates for each vertex in sequential order. The JavaScript code then applies the Shoelace Formula to compute the area and displays the result.
Use Cases:
Surveying: Determining the area of land parcels based on boundary coordinates.
Computer Graphics: Calculating the area of shapes for rendering, collision detection, or texture mapping.
Engineering and Architecture: Estimating the floor space of irregularly shaped rooms or structures.
Urban Planning: Measuring the area of city blocks or development zones.
Education: A practical tool for students learning geometry and coordinate systems.
Remember to input your vertices in sequential order (either clockwise or counterclockwise) to ensure an accurate calculation.
function updateVertexInputs() {
var numVertices = parseInt(document.getElementById("vertices").value);
var vertexInputsDiv = document.getElementById("vertexInputs");
vertexInputsDiv.innerHTML = ""; // Clear previous inputs
if (isNaN(numVertices) || numVertices < 3) {
numVertices = 3;
document.getElementById("vertices").value = 3;
}
for (var i = 0; i < numVertices; i++) {
var vertexDiv = document.createElement("div");
vertexDiv.style.display = "flex";
vertexDiv.style.flexWrap = "wrap";
vertexDiv.style.gap = "10px";
vertexDiv.style.alignItems = "center";
vertexDiv.style.marginBottom = "10px";
var label = document.createElement("label");
label.textContent = "Vertex " + (i + 1) + " (x, y):";
label.style.flex = "1 1 150px";
label.style.fontWeight = "600";
label.style.color = "#555";
var xInput = document.createElement("input");
xInput.type = "number";
xInput.className = "vertex-x";
xInput.placeholder = "X" + (i + 1);
xInput.style.flex = "0 1 100px";
xInput.style.padding = "10px 12px";
xInput.style.border = "1px solid #dee2e6";
xInput.style.borderRadius = "4px";
xInput.style.fontSize = "1rem";
xInput.style.boxSizing = "border-box";
var yInput = document.createElement("input");
yInput.type = "number";
yInput.className = "vertex-y";
yInput.placeholder = "Y" + (i + 1);
yInput.style.flex = "0 1 100px";
yInput.style.padding = "10px 12px";
yInput.style.border = "1px solid #dee2e6";
yInput.style.borderRadius = "4px";
yInput.style.fontSize = "1rem";
yInput.style.boxSizing = "border-box";
vertexDiv.appendChild(label);
vertexDiv.appendChild(xInput);
vertexDiv.appendChild(yInput);
vertexInputsDiv.appendChild(vertexDiv);
}
}
function calculatePolygonArea() {
var xCoords = [];
var yCoords = [];
var vertexInputs = document.querySelectorAll("#vertexInputs .vertex-x, #vertexInputs .vertex-y");
var resultDiv = document.getElementById("result");
var isValid = true;
vertexInputs.forEach(function(input) {
var value = parseFloat(input.value);
if (isNaN(value)) {
isValid = false;
}
if (input.classList.contains("vertex-x")) {
xCoords.push(value);
} else {
yCoords.push(value);
}
});
if (!isValid || xCoords.length < 3 || yCoords.length < 3 || xCoords.length !== yCoords.length) {
resultDiv.textContent = "Invalid Input";
resultDiv.style.backgroundColor = "#dc3545"; // Error color
return;
}
var numVertices = xCoords.length;
var sum1 = 0;
var sum2 = 0;
for (var i = 0; i < numVertices; i++) {
var j = (i + 1) % numVertices; // Next vertex index, wraps around
sum1 += xCoords[i] * yCoords[j];
sum2 += yCoords[i] * xCoords[j];
}
var area = 0.5 * Math.abs(sum1 – sum2);
resultDiv.textContent = "Area: " + area.toFixed(4);
resultDiv.style.backgroundColor = "var(–success-green)"; // Reset to success color
}
// Initialize input fields on page load
document.addEventListener("DOMContentLoaded", updateVertexInputs);