Math

Equation of a Circle Homework Help

The equation of a circle describes every point that sits an equal distance (the radius) from one fixed point (the center). Most textbook problems either give the center and radius and ask for the equation, or give the equation and ask to identify the center, radius, or whether a point lies on the circle.

Common topics

Standard form

A circle with center (h, k) and radius r has equation (x - h)² + (y - k)² = r². When the center is the origin (0, 0), this simplifies to x² + y² = r².

Reading off the center and radius

Given an equation already in standard form, the center is (h, k) — note the SIGN FLIP, since the formula subtracts h and k — and the radius is the square root of the number on the right side.

Completing the square

If the equation is given in expanded form (x² + y² + Dx + Ey + F = 0), group the x-terms and y-terms separately and complete the square on each to convert it back to standard form.

Testing whether a point lies on the circle

Substitute the point's coordinates into the equation — if both sides are equal, the point lies exactly on the circle; if the left side is less than r², the point is inside; if greater, the point is outside.

Tangent lines and the radius

A line tangent to a circle touches it at exactly one point, and at that point the tangent line is always perpendicular to the radius drawn to it — this perpendicularity is the key fact used to find a tangent line's equation.

Worked example

Find the equation of a circle with center (0, 0) that passes through the point (3, 4).

The radius is the distance from the center to the given point on the circle.
Since the center is the origin, use the distance formula: r = √(3² + 4²) = √(9 + 16) = √25 = 5.
Substitute the center (0, 0) and radius 5 into the standard form (x - h)² + (y - k)² = r².
This gives x² + y² = 25.

Answer: x² + y² = 25 (center at the origin, radius 5).

Get help with Equation of a Circle

AI Solve Quiz walks through the same center-radius identification and completing-the-square process shown above for a photographed circle-equation problem, showing every intermediate line rather than jumping straight to the final equation.

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Frequently asked questions

Why is the center written with a minus sign in the formula? ▼

The formula (x - h)² + (y - k)² = r² comes directly from the distance formula between a point (x, y) and the center (h, k). So if the equation shows (x - 3)², the center's x-coordinate is +3, not -3 — the equation subtracts the center coordinate.

How do I know if the equation is even a circle? ▼

After completing the square, the right-hand side must be a positive number for the equation to represent a real circle. If it equals zero, the "circle" is really just a single point; if it is negative, no real circle exists.

Is the number on the right side of the equation the radius? ▼

No — it is the radius SQUARED. A very common mistake is reading (x - h)² + (y - k)² = 25 and writing the radius as 25 instead of taking the square root to get 5. Always take the square root of that final number before stating the radius.

What is the relationship between a circle's equation and the distance formula? ▼

The circle equation IS the distance formula, just rearranged. The distance between any point (x, y) and the center (h, k) is √((x-h)² + (y-k)²). Setting that distance equal to a fixed radius r and squaring both sides to remove the square root produces exactly (x - h)² + (y - k)² = r² — every point satisfying this equation is, by definition, exactly r units from the center.

Related subjects

Linear Equations A linear equation is any equation where the variable appears only to the first power — no squares, no roots, no variable in a denominator on its own. Textbook homework usually mixes fractions, parentheses, and variables on both sides, which is where most mistakes creep in even though the underlying method never changes. Angle Relationships in Parallel Lines When a straight line (a transversal) crosses two parallel lines, it creates eight angles in total, and geometry homework usually asks students to name and count specific pairs among them — a frequent source of confusion because several angle-pair names sound alike. Graphing Quadratic Functions A quadratic function y = ax² + bx + c graphs as a parabola, and most homework questions ask for specific features of that curve — its vertex, its axis of symmetry, or which way it opens — rather than a full point-by-point plot. Verb Tense Exercises Verb-tense fill-in-the-blank exercises give a sentence with a verb in brackets and ask the student to write the correct form — the right choice almost always comes from a time clue elsewhere in the sentence, not from memorizing the verb in isolation.

This page explains a common homework pattern for study purposes. Always follow your school's academic integrity policy.