Understanding the Equation of a Circle
In geometry, a circle is defined as the set of all points in a plane that are at a fixed distance, called the radius, from a fixed point, called the center. Expressing this relationship algebraically allows us to solve complex coordinate geometry problems.
The Standard Form Equation
The standard form of a circle's equation is the most intuitive way to represent it because it clearly displays the center (h, k) and the radius (r). The formula is:
(x – h)² + (y – k)² = r²
- h: The x-coordinate of the center.
- k: The y-coordinate of the center.
- r: The radius of the circle.
The General Form Equation
By expanding the standard form, we arrive at the general form of the circle equation:
x² + y² + Dx + Ey + F = 0
To convert from standard form to general form, we use these relationships:
- D = -2h
- E = -2k
- F = h² + k² – r²
Practical Example
Suppose you have a circle with a center at (3, -4) and a radius of 6.
- Identify h, k, and r: h = 3, k = -4, r = 6.
- Standard Form: Plug the values in: (x – 3)² + (y – (-4))² = 6², which simplifies to (x – 3)² + (y + 4)² = 36.
- General Form Calculation:
- D = -2(3) = -6
- E = -2(-4) = 8
- F = 3² + (-4)² – 6² = 9 + 16 – 36 = -11
The general equation is x² + y² – 6x + 8y – 11 = 0.
Properties of a Circle
Beyond the equation, two fundamental properties are often calculated:
- Area: πr² (The total space inside the circle).
- Circumference: 2πr (The distance around the outside of the circle).
function calculateCircleEquation() {
var h = parseFloat(document.getElementById('centerX').value);
var k = parseFloat(document.getElementById('centerY').value);
var r = parseFloat(document.getElementById('radius').value);
if (isNaN(h) || isNaN(k) || isNaN(r)) {
alert("Please enter valid numeric values.");
return;
}
if (r = 0 ? "(x – " + h + ")" : "(x + " + Math.abs(h) + ")";
var kPart = k >= 0 ? "(y – " + k + ")" : "(y + " + Math.abs(k) + ")";
var rSquared = r * r;
// Clean up 0 values
if (h === 0) hPart = "x²"; else hPart = hPart + "²";
if (k === 0) kPart = "y²"; else kPart = kPart + "²";
var standardStr = hPart + " + " + kPart + " = " + rSquared.toFixed(2).replace(/\.00$/, ");
// 2. General Form Construction: x² + y² + Dx + Ey + F = 0
var d = -2 * h;
var e = -2 * k;
var f = (h * h) + (k * k) – (r * r);
var generalStr = "x² + y²";
if (d !== 0) {
generalStr += (d > 0 ? " + " + d : " – " + Math.abs(d)) + "x";
}
if (e !== 0) {
generalStr += (e > 0 ? " + " + e : " – " + Math.abs(e)) + "y";
}
if (f !== 0) {
generalStr += (f > 0 ? " + " + f.toFixed(2).replace(/\.00$/, ") : " – " + Math.abs(f).toFixed(2).replace(/\.00$/, "));
}
generalStr += " = 0″;
// 3. Metrics
var area = Math.PI * r * r;
var circumference = 2 * Math.PI * r;
// Display Results
document.getElementById('standardForm').innerHTML = standardStr;
document.getElementById('generalForm').innerHTML = generalStr;
document.getElementById('circleArea').innerText = area.toFixed(4);
document.getElementById('circleCircum').innerText = circumference.toFixed(4);
document.getElementById('results').style.display = 'block';
}