Equation of a Circle: Standard & General Form

TL;DR

The equation of a circle in standard form is (x − h)² + (y − k)² = r², where (h, k) is the centre and r is the radius. This article covers the standard and general forms, the derivation straight from the Pythagorean theorem, how to read the centre and radius off either form, how to convert between them by completing the square, and six worked examples.

What Is the Equation of a Circle?

The equation of a circle is the algebraic rule that every point on the circle satisfies, and no other point does. Because a circle is the set of points a fixed distance r from a centre (h,k), the equation simply says "the distance from (x,y) to the centre equals r."

In its cleanest standard form, the equation of a circle with centre (h,k) and radius r is:

(x−h)² + (y−k)² = r².

When the centre sits at the origin (0,0), the equation collapses to its simplest version:

x² + y² = r².

Deriving the Equation from the Pythagorean Theorem

The equation is derived from the Pythagorean theorem and the distance formula. Take any point (x,y) on a circle centred at (h,k). Drop a horizontal line and a vertical line to form a right triangle with the radius as the hypotenuse:

• The horizontal leg has length x−h.
• The vertical leg has length y−k.
• The hypotenuse is the radius, r.

The Pythagorean theorem states that hypotenuse squared equals the sum of the squares of the legs:

(x−h)² + (y−k)² = r².

That is the standard form, derived in one line. The distance formula is the same statement: the distance from (x,y) to (h,k) is √((x−h)² + (y−k)²), and setting that distance equal to r and squaring both sides gives the equation again.

The General Form of the Equation of a Circle

Expand the standard form to get a second equivalent way of writing the same circle, called the general form:

x² + y² + 2gx + 2fy + c = 0.

Here the constants relate to the centre and radius by:

centre = (−g,−f), r = √(g² + f² − c).

How to Find the Centre and Radius — Completing the Square

To pull the centre and radius out of a general-form equation, complete the square on the x-terms and the y-terms separately. The move is always the same: take half the coefficient of x, square it, and add it (and do the same for y), balancing the equation as you go.

Examples of the Equation of a Circle

Example 1 - Find the centre and radius of the circle (x−4)² + (y+7)² = 9

Match against (x−h)² + (y−k)² = r². Here h = 4, and since (y+7) = (y−(−7)), we have k = −7. The right side is r² = 9, so r = 3.

Final answer: centre (4,−7), radius 3.

Example 2 - Write the equation of a circle with centre (−2,5) and radius 6

The correct equation is:

(x+2)² + (y−5)² = 36.

Example 3 - Write the equation of a circle centred at the origin with radius 10

The equation is:

x² + y² = 100.

Example 4 - Find the centre and radius of the circle x² + y² − 6x + 4y − 12 = 0

Group and complete the square on x and y separately:

(x − 3)² + (y + 2)² = 25.

Final answer: centre (3,−2), radius 5.

Example 5 - Find the centre and radius of x² + y² + 8x − 10y + 5 = 0 using the general-form shortcut

Final answer: centre (−4,5), radius 6.

Example 6 - A circle has a diameter with endpoints A(1,2) and B(7,10). Find its equation

The equation is:

(x−4)² + (y−6)² = 25.

Where the Equation of a Circle Shows Up

The equation matters because it turns a circle into something a computer or an engineer can calculate with:

• GPS and trilateration.
• Computer graphics and games.
• Engineering and CAD.
• Conic sections and orbits.

Where Students Trip Up on the Equation of a Circle

Mistake 1: Getting the sign wrong on the centre

Mistake 2: Writing r instead of r² on the right.

Mistake 3: Forgetting to balance both sides when completing the square.

Key Takeaways

• The equation of a circle is (x−h)² + (y−k)² = r².
• Derived from the Pythagorean theorem.
• The general form hides the centre and radius.
• Complete the square to convert general form to standard form.
• Common mistakes include flipping the centre's sign and writing r instead of r².

Practice These Problems to Solidify Your Understanding

1. Find the centre and radius of (x−5)² + (y+3)² = 49.
2. Write the equation of a circle with centre (0,−4) and radius 8.
3. Find the centre and radius of x² + y² − 4x + 6y − 3 = 0.

Answer to Question 1: centre (5,−3), radius 7. Answer to Question 2: x² + (y + 4)² = 64. Answer to Question 3: centre (2,−3), radius 4.

Frequently Asked Questions

1. What is the standard equation of a circle? (x−h)² + (y−k)² = r².
2. What is the general form of the equation of a circle? x² + y² + 2gx + 2fy + c = 0.
3. How do you find the centre and radius from a general-form equation? Complete the square or use the shortcut.
4. How do you know if an equation represents a circle? Check for equal x² and y² coefficients.
5. What is C in the general equation of a circle? The constant term, linked to the radius.