Side Length (s)
Apothem (a) and Side (s)
Circumradius (R)
Resulting Area
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How to Calculate the Area of a Regular Pentagon
A regular pentagon is a five-sided polygon where all sides and internal angles are equal. Calculating its area depends on which measurements you have available. There are three primary methods used in geometry.
1. Using the Side Length
If you only know the length of one side (s), the formula is derived from the geometry of five internal triangles:
The apothem (a) is the distance from the center of the pentagon to the midpoint of any side. Since the pentagon can be divided into 5 equal triangles, the area is:
Area = (1/2) × Perimeter × Apothem = (5/2) × s × a
3. Using the Circumradius
The circumradius (R) is the distance from the center to any vertex. The formula using trigonometry is:
Area = (5/2) × R² × sin(72°) ≈ 2.3776 × R²
Example Calculation
Suppose you have a regular pentagon with a side length of 10 cm.
Identify the side (s) = 10.
Apply the side formula: Area ≈ 1.7204774 × 10².
Area ≈ 1.7204774 × 100.
Total Area ≈ 172.05 cm².
function toggleInputs() {
var method = document.getElementById('calcMethod').value;
var sideGroup = document.getElementById('sideInputGroup');
var apothemGroup = document.getElementById('apothemInputGroup');
var radiusGroup = document.getElementById('radiusInputGroup');
sideGroup.style.display = 'none';
apothemGroup.style.display = 'none';
radiusGroup.style.display = 'none';
if (method === 'side') {
sideGroup.style.display = 'block';
} else if (method === 'apothem') {
sideGroup.style.display = 'block';
apothemGroup.style.display = 'block';
} else if (method === 'circumradius') {
radiusGroup.style.display = 'block';
}
document.getElementById('pentagonResult').style.display = 'none';
}
function calculatePentagonArea() {
var method = document.getElementById('calcMethod').value;
var area = 0;
var formulaTxt = "";
if (method === 'side') {
var s = parseFloat(document.getElementById('sideLength').value);
if (isNaN(s) || s <= 0) {
alert("Please enter a valid side length.");
return;
}
// Formula: Area = 1/4 * sqrt(5(5+2*sqrt(5))) * s^2
var multiplier = 0.25 * Math.sqrt(5 * (5 + 2 * Math.sqrt(5)));
area = multiplier * Math.pow(s, 2);
formulaTxt = "Formula: (1/4)√[5(5+2√5)] × s²";
}
else if (method === 'apothem') {
var s = parseFloat(document.getElementById('sideLength').value);
var a = parseFloat(document.getElementById('apothemLength').value);
if (isNaN(s) || s <= 0 || isNaN(a) || a <= 0) {
alert("Please enter valid side and apothem lengths.");
return;
}
// Formula: Area = 1/2 * Perimeter * Apothem = 1/2 * (5*s) * a
area = 0.5 * (5 * s) * a;
formulaTxt = "Formula: (5/2) × s × a";
}
else if (method === 'circumradius') {
var R = parseFloat(document.getElementById('radiusLength').value);
if (isNaN(R) || R <= 0) {
alert("Please enter a valid radius.");
return;
}
// Formula: Area = 5/2 * R^2 * sin(72 deg)
// 72 degrees in radians is 72 * PI / 180
var sin72 = Math.sin(72 * Math.PI / 180);
area = (5 / 2) * Math.pow(R, 2) * sin72;
formulaTxt = "Formula: (5/2) × R² × sin(72°)";
}
var resultDiv = document.getElementById('pentagonResult');
var areaDisplay = document.getElementById('areaValue');
var formulaDisplay = document.getElementById('formulaUsed');
areaDisplay.innerText = area.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 4});
formulaDisplay.innerText = formulaTxt;
resultDiv.style.display = 'block';
}