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Derive the
Equation of a Circle

Learn how the Pythagorean Theorem connects to every circle, and master completing the square to reveal center and radius from any equation.

Geometry

What is a Circle, Really?

A circle is the set of all points in a plane that are a fixed distance from a single point. That fixed distance is the radius \( r \), and that single point is the center \( (h, k) \).

\( (h, k) \)
\( r \)
\( (x, y) \)
same \( r \)

From Distance to Equation

Pick any point \( (x, y) \) on the circle with center \( (h, k) \). Drop a horizontal and vertical segment to form a right triangle. The legs are \( |x - h| \) and \( |y - k| \), and the hypotenuse is \( r \).

\( (h, k) \)
\( (x, y) \)
\( |x - h| \)
\( |y - k| \)
\( r \)

The Standard Form Equation

The equation of a circle with center \( (h, k) \) and radius \( r \) is:

\( h \)
The \( x \)-coordinate of the center
\( k \)
The \( y \)-coordinate of the center
\( r \)
The radius (always positive)

Special Case: Center at the Origin

When the center is \( (0, 0) \), both \( h \) and \( k \) are zero, so the equation simplifies beautifully:

\( x \)
\( y \)
\( (0, 0) \)
\( r \)
\( (r, 0) \)

This is the simplest circle equation: just \( x^2 + y^2 = r^2 \).

Standard vs General Form

Standard Form General Form
Equation \( (x-h)^2 + (y-k)^2 = r^2 \) \( x^2 + y^2 + Dx + Ey + F = 0 \)
Center Read directly: \( (h, k) \) Must complete the square
Radius Read directly: \( r \) Must complete the square
Best for Graphing, identifying parts Algebraic manipulation

What is Completing the Square?

Completing the square converts \( x^2 + bx \) into a perfect square trinomial. The idea: add and subtract \( \left(\frac{b}{2}\right)^2 \) so you can factor into \( \left(x + \frac{b}{2}\right)^2 \).

Example: \( x^2 + 6x \). Half of 6 is 3; square it to get 9. So \( x^2 + 6x + 9 = (x + 3)^2 \).

How to Convert General to Standard

1
Group \( x \)-terms and \( y \)-terms together. Move the constant to the right side.
2
Complete the square for the \( x \)-group: add \( \left(\frac{b_x}{2}\right)^2 \) to both sides.
3
Complete the square for the \( y \)-group: add \( \left(\frac{b_y}{2}\right)^2 \) to both sides.
4
Factor each perfect square trinomial and simplify the right side.
5
Read off center \( (h, k) \) and radius \( r = \sqrt{\text{right side}} \).

Example 1: Write the Equation

Write the equation of a circle with center \( (3, -2) \) and radius \( r = 5 \).

\( x \)
\( y \)
\( (3, -2) \)
\( r = 5 \)
\( 0 \)
\( 3 \)
\( -2 \)

Example 2: Complete the Square

Find the center and radius of: \( x^2 + y^2 - 6x + 4y - 12 = 0 \)

1
Group and move constant: \( (x^2 - 6x) + (y^2 + 4y) = 12 \)
2
Complete \( x \)-square: half of \( -6 \) is \( -3 \), squared is \( 9 \). Add \( 9 \) to both sides.
3
Complete \( y \)-square: half of \( 4 \) is \( 2 \), squared is \( 4 \). Add \( 4 \) to both sides.
4
Factor: \( (x - 3)^2 + (y + 2)^2 = 12 + 9 + 4 = 25 \)

Example 3: Trickier Coefficients

Find center and radius: \( x^2 + y^2 + 10x - 2y + 17 = 0 \)

1
Group: \( (x^2 + 10x) + (y^2 - 2y) = -17 \)
2
Complete \( x \): \( \left(\frac{10}{2}\right)^2 = 25 \). Add 25 to both sides.
3
Complete \( y \): \( \left(\frac{-2}{2}\right)^2 = 1 \). Add 1 to both sides.
4
Factor: \( (x + 5)^2 + (y - 1)^2 = -17 + 25 + 1 = 9 \)

Graphing from Standard Form

Graph \( (x + 5)^2 + (y - 1)^2 = 9 \) from Example 3.

\( x \)
\( y \)
\( 0 \)
\( (-5, 1) \)
\( r = 3 \)
\( -5 \)
\( -2 \)

Why Does Completing the Square Work?

When we expand standard form, we get general form. Completing the square reverses that expansion. Watch:

→
Standard: \( (x - 3)^2 + (y + 2)^2 = 25 \)
→
Expand: \( x^2 - 6x + 9 + y^2 + 4y + 4 = 25 \)
→
General: \( x^2 + y^2 - 6x + 4y - 12 = 0 \)

Watch the Signs Carefully

The standard form always uses subtraction: \( (x - h)^2 + (y - k)^2 = r^2 \). This means the sign inside the parentheses is opposite to the coordinate of the center.

\( (x - 3)^2 \)
Center \( x \)-value is \( +3 \)
\( (x + 4)^2 \)
Center \( x \)-value is \( -4 \) because \( (x + 4) = (x - (-4)) \)

Common Mistakes to Avoid

✕ Forgetting to Add to Both Sides
When completing the square, every value you add to the left must also be added to the right to keep the equation balanced.
✕ Sign Error on the Center
Reading \( (x + 5)^2 \) as center \( x = 5 \) instead of \( x = -5 \). The standard form uses subtraction, so flip the sign.
✕ Confusing \( r \) with \( r^2 \)
The right side of the equation is \( r^2 \), not \( r \). If the right side is 25, the radius is \( \sqrt{25} = 5 \), not 25.

Question 1 of 5

What is the equation of a circle with center \( (2, -7) \) and radius \( r = 4 \)?

Question 2 of 5

What is the center and radius of the circle \( (x + 1)^2 + (y - 5)^2 = 49 \)?

Question 3 of 5

Complete the square to find the radius of: \( x^2 + y^2 - 8x + 6y = 0 \)

Question 4 of 5

Which equation represents a circle centered at the origin with radius \( \sqrt{13} \)?

Question 5 of 5

What is the center of the circle given by \( x^2 + y^2 + 12x - 4y + 15 = 0 \)?