Circles
Understanding the geometric blueprint of a circle and its algebraic representation.
📐 The Definition of a Circle
Mathematically, a circle is not just a "round shape." It is the set of all points \((x, y)\) in a plane that are at a fixed distance, called the radius (\(r\)), from a fixed point, called the center \((h, k)\).
The Distance Formula Proof
To find the equation of a circle, we simply apply the Distance Formula between the center \((h, k)\) and any point \((x, y)\) on the edge:
By squaring both sides, we arrive at the Standard Form:
The Pythagorean Connection
Notice anything familiar? The standard form is actually just the Pythagorean Theorem (\(a^2 + b^2 = c^2\)) rewritten for a coordinate grid. The horizontal distance is \((x-h)\), the vertical distance is \((y-k)\), and the hypotenuse is the radius \(r\).
Next Steps in Your Training
Now that you understand why the equation works, learn how to solve it fast.