Equation of a Circle Calculator
:root {
–primary-blue: #004a99;
–success-green: #28a745;
–light-background: #f8f9fa;
–dark-text: #333;
–border-color: #ccc;
}
body {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
background-color: var(–light-background);
color: var(–dark-text);
line-height: 1.6;
margin: 0;
padding: 20px;
}
.calculator-container {
max-width: 800px;
margin: 40px auto;
background-color: #ffffff;
padding: 30px;
border-radius: 8px;
box-shadow: 0 4px 15px rgba(0, 0, 0, 0.1);
border: 1px solid var(–border-color);
}
h1, h2 {
color: var(–primary-blue);
text-align: center;
margin-bottom: 20px;
}
.input-group {
margin-bottom: 20px;
padding: 15px;
border: 1px solid var(–border-color);
border-radius: 5px;
background-color: #fdfdfd;
display: flex;
align-items: center;
flex-wrap: wrap; /* Allow wrapping on smaller screens */
}
.input-group label {
display: inline-block;
width: 180px; /* Fixed width for labels */
font-weight: bold;
margin-right: 15px;
color: var(–primary-blue);
flex-shrink: 0; /* Prevent label shrinking */
}
.input-group input[type="number"] {
flex-grow: 1; /* Allow input to take available space */
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
font-size: 1rem;
min-width: 150px; /* Minimum width for inputs */
box-sizing: border-box; /* Include padding and border in element's total width */
}
.input-group input[type="number"]:focus {
outline: none;
border-color: var(–primary-blue);
box-shadow: 0 0 0 2px rgba(0, 74, 153, 0.2);
}
button {
display: block;
width: 100%;
padding: 12px 20px;
background-color: var(–primary-blue);
color: white;
border: none;
border-radius: 5px;
font-size: 1.1rem;
font-weight: bold;
cursor: pointer;
transition: background-color 0.3s ease;
margin-top: 10px;
}
button:hover {
background-color: #003366;
}
.result-section {
margin-top: 30px;
padding: 25px;
background-color: var(–light-background);
border: 1px solid var(–border-color);
border-radius: 5px;
text-align: center;
}
.result-section h2 {
margin-top: 0;
color: var(–primary-blue);
}
#equationResult {
font-size: 1.8rem;
font-weight: bold;
color: var(–success-green);
word-break: break-all; /* Ensure long equations wrap */
padding: 15px;
background-color: #e9ecef;
border-radius: 5px;
min-height: 50px; /* Ensure minimum height */
display: flex;
align-items: center;
justify-content: center;
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margin-top: 50px;
padding: 30px;
background-color: #ffffff;
border-radius: 8px;
box-shadow: 0 4px 15px rgba(0, 0, 0, 0.1);
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}
.article-section h2 {
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margin-bottom: 20px;
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.article-section p,
.article-section ul,
.article-section li {
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color: #555;
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.article-section ul {
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padding-left: 20px;
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.article-section code {
background-color: #e9ecef;
padding: 2px 5px;
border-radius: 3px;
font-family: Consolas, Monaco, 'Andale Mono', 'Ubuntu Mono', monospace;
}
/* Responsive adjustments */
@media (max-width: 768px) {
.input-group {
flex-direction: column;
align-items: flex-start;
}
.input-group label {
width: auto; /* Auto width for labels on smaller screens */
margin-bottom: 10px;
}
.input-group input[type="number"] {
width: 100%; /* Full width for inputs */
}
}
Equation of a Circle Calculator
Standard Equation of the Circle
Enter values to see the equation.
Understanding the Equation of a Circle
The equation of a circle is a fundamental concept in coordinate geometry that describes the set of all points equidistant from a central point. This distance is known as the radius, and the central point is the center of the circle.
The Standard Form
The most common and intuitive form of a circle's equation is the standard form. If a circle has its center at coordinates (h, k) and a radius of length r, then the equation is given by:
(x - h)² + (y - k)² = r²
In this equation:
(x, y) represents any point on the circumference of the circle.h is the x-coordinate of the circle's center.k is the y-coordinate of the circle's center.r is the radius of the circle.
How the Calculator Works
This calculator takes the coordinates of the circle's center (h and k) and its radius (r) as input. It then directly substitutes these values into the standard form equation.
For example, if you input:
- Center X (h) =
2
- Center Y (k) =
-3
- Radius (r) =
5
The calculator will compute:
h = 2k = -3r² = 5² = 25
Substituting these into the standard equation (x - h)² + (y - k)² = r² yields:
(x - 2)² + (y - (-3))² = 25
Which simplifies to:
(x - 2)² + (y + 3)² = 25
This resulting equation precisely defines the circle with the given center and radius.
Use Cases for the Equation of a Circle
Understanding and calculating the equation of a circle has various applications:
- Mathematics Education: Essential for learning coordinate geometry, conic sections, and graphing.
- Computer Graphics: Used to draw circles, arcs, and in collision detection algorithms in games and simulations.
- Engineering and Physics: Modeling circular motion, designing circular structures, and analyzing wave propagation.
- Astronomy: Describing orbits or celestial bodies under certain simplified conditions.
- Navigation Systems: Calculating ranges and locations.
By accurately defining a circle's position and size through its equation, we can perform complex geometric calculations and model real-world phenomena.
function calculateEquation() {
var centerXInput = document.getElementById("centerX");
var centerYInput = document.getElementById("centerY");
var radiusInput = document.getElementById("radius");
var equationResultDiv = document.getElementById("equationResult");
var h = parseFloat(centerXInput.value);
var k = parseFloat(centerYInput.value);
var r = parseFloat(radiusInput.value);
// Clear previous error messages or results
equationResultDiv.style.color = "var(–success-green)";
// Input validation
if (isNaN(h) || isNaN(k) || isNaN(r)) {
equationResultDiv.innerHTML = "Please enter valid numbers for all fields.";
equationResultDiv.style.color = "red";
return;
}
if (r 0) {
equationString += "(x – " + h + ")²";
} else { // h 0) {
equationString += "(y – " + k + ")²";
} else { // k < 0
equationString += "(y + " + Math.abs(k) + ")²";
}
equationString += " = " + rSquared;
equationResultDiv.innerHTML = equationString;
}