The Equation of a Circle

A circle can be described geometrically as all points in a plane that are the same distance from a fixed point.

The fixed point is the center, and the fixed distance is the radius.

On a coordinate plane, this definition can be represented algebraically using the equation of a circle. The equation tells us exactly where the circle is located and how large it is.

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Standard Form of a Circle Equation

The standard form of the equation of a circle is:

(x − h)² + (y − k)² = r²

where:

• (h, k) is the center
• r is the radius
• r² is the square of the radius
• (x, y) represents any point on the circle

This equation is one of the most useful ways to describe a circle on a coordinate plane.

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Understanding the Equation

Consider:

(x − 3)² + (y − 2)² = 16

Compare this with:

(x − h)² + (y − k)² = r²

We can identify:

h = 3

k = 2

r² = 16

Therefore:

r = 4

The circle has:

center = (3, 2)

radius = 4

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Why the Circle Equation Works

The equation of a circle comes from the distance formula, which itself comes from the Pythagorean theorem.

Suppose:

• the center is (h, k)
• a point on the circle is (x, y)

The horizontal distance between the points is:

x − h

The vertical distance is:

y − k

Because the distance from the center to every point on the circle is r:

(x − h)² + (y − k)² = r²

This is exactly the standard equation of a circle.

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Recognizing a Circle Equation

An equation such as:

(x − 4)² + (y + 1)² = 25

is in standard circle form because it follows the pattern:

(x − h)² + (y − k)² = r²

Both variable expressions are squared and added together, and the right side represents the square of the radius.

From the equation we can determine the center and radius immediately.

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Reading the Center Carefully

One of the most important skills is reading the signs correctly.

Consider:

(x − 5)² + (y − 3)² = 49

The center is:

(5, 3)

Now consider:

(x + 5)² + (y − 3)² = 49

Rewrite mentally:

x + 5 = x − (−5)

Therefore, the center is:

(−5, 3)

The sign inside each bracket is the opposite of the corresponding center coordinate.

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The Sign Rule

If the equation contains:

(x − 4)²

then:

h = 4

If it contains:

(x + 4)²

then:

h = −4

Similarly:

(y − 7)² → k = 7

(y + 7)² → k = −7

A useful reminder is:

The signs inside the brackets are opposite the coordinates of the center.

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Worked Example: Identify Center and Radius

Consider:

(x − 2)² + (y + 6)² = 36

Compare with:

(x − h)² + (y − k)² = r²

For x:

h = 2

For y:

y + 6 = y − (−6)

so:

k = −6

For the radius:

r² = 36

r = √36

r = 6

Therefore:

center = (2, −6)

radius = 6

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The Radius Is Not r²

Consider:

(x + 1)² + (y − 4)² = 81

The number on the right is:

r² = 81

It is not the radius itself.

Take the square root:

r = √81

r = 9

Therefore:

center = (−1, 4)

radius = 9

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Worked Example with a Non-Perfect Square

Consider:

(x − 3)² + (y + 2)² = 20

The center is:

(3, −2)

and:

r² = 20

Therefore:

r = √20

Simplify:

r = √(4 × 5)

r = 2√5

An approximate decimal value is:

r ≈ 4.47

The radius does not have to be a whole number.

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Circles Centered at the Origin

The origin is:

(0, 0)

If a circle has its center at the origin, then:

h = 0

k = 0

The standard equation becomes:

x² + y² = r²

For example:

x² + y² = 25

has:

center = (0, 0)

radius = 5

This is an important special case.

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Writing an Equation from the Center and Radius

Suppose a circle has:

center = (4, 3)

radius = 5

Start with:

(x − h)² + (y − k)² = r²

Substitute:

h = 4

k = 3

r = 5

Therefore:

(x − 4)² + (y − 3)² = 25

Notice that the right side contains:

5² = 25

not simply 5.

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Worked Example with a Negative Coordinate

Write the equation of a circle with:

center = (−3, 2)

radius = 4

Start with:

(x − h)² + (y − k)² = r²

Substitute:

(x − (−3))² + (y − 2)² = 4²

Simplify:

(x + 3)² + (y − 2)² = 16

The negative x-coordinate produces a plus sign inside the bracket.

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Worked Example with Two Negative Coordinates

Write the equation for a circle with:

center = (−5, −2)

radius = 7

Substitute:

(x − (−5))² + (y − (−2))² = 7²

Simplify:

(x + 5)² + (y + 2)² = 49

This is why careful use of parentheses is helpful when the center contains negative coordinates.

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Worked Example Centered at the Origin

Suppose:

center = (0, 0)

radius = 8

Then:

(x − 0)² + (y − 0)² = 8²

Therefore:

x² + y² = 64

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Writing an Equation from the Diameter

Sometimes the diameter is given instead of the radius.

Suppose a circle has:

center = (2, 5)

diameter = 12

First find the radius:

r = d/2

r = 12/2

r = 6

Now:

r² = 36

Therefore:

(x − 2)² + (y − 5)² = 36

Always convert diameter to radius before writing the equation.

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Writing an Equation from a Center and a Point

Sometimes the radius is not given directly.

Suppose a circle has center:

(1, 2)

and passes through:

(4, 6)

The radius is the distance between these points.

Use the distance relationship:

r² = (4 − 1)² + (6 − 2)²

r² = 3² + 4²

r² = 9 + 16

r² = 25

Therefore:

r = 5

and the circle equation is:

(x − 1)² + (y − 2)² = 25

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Verifying Whether a Point Lies on a Circle

A point lies on a circle if its coordinates make the circle equation true.

Suppose:

(x − 2)² + (y − 1)² = 25

Does:

(5, 5)

lie on the circle?

Substitute:

x = 5

y = 5

(5 − 2)² + (5 − 1)² = 25

3² + 4² = 25

9 + 16 = 25

25 = 25

The equation is true.

Therefore:

(5, 5) lies on the circle.

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A Point That Does Not Lie on the Circle

Use the same circle:

(x − 2)² + (y − 1)² = 25

Test:

(4, 3)

Substitute:

(4 − 2)² + (3 − 1)²

= 2² + 2²

= 4 + 4

= 8

But:

8 ≠ 25

Therefore:

(4, 3) does not lie on the circle.

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Inside, On, or Outside the Circle

Substitution can tell us more than whether a point lies exactly on the circle.

For:

(x − h)² + (y − k)² = r²

calculate the left side for the point.

If:

left side = r²

the point is on the circle.

If:

left side < r²

the point is inside the circle.

If:

left side > r²

the point is outside the circle.

This works because the left side represents the square of the point's distance from the center.

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Worked Example: Inside, On, or Outside

Consider:

x² + y² = 25

Test:

(2, 3)

Calculate:

2² + 3² = 4 + 9 = 13

Since:

13 < 25

the point is:

inside the circle

Now test:

(3, 4)

3² + 4² = 9 + 16 = 25

Therefore:

(3, 4) is on the circle

Now test:

(5, 4)

5² + 4² = 25 + 16 = 41

Since:

41 > 25

the point is:

outside the circle

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Graphing a Circle from Its Equation

Suppose we want to graph:

(x − 2)² + (y − 3)² = 16

First identify:

center = (2, 3)

radius = 4

Plot the center:

(2, 3)

Then move 4 units:

• right
• left
• up
• down

from the center.

This gives four useful points on the circle.

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Finding the Four Main Points

For center:

(h, k)

and radius:

r

the four main points are:

Right: (h + r, k)

Left: (h − r, k)

Top: (h, k + r)

Bottom: (h, k − r)

For:

center = (2, 3)

radius = 4

we obtain:

Right: (6, 3)

Left: (−2, 3)

Top: (2, 7)

Bottom: (2, −1)

These points help us sketch the circle accurately.

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Graphing Example

Graph:

(x + 1)² + (y − 2)² = 9

Identify the center:

(−1, 2)

Find the radius:

r = √9

r = 3

Main points:

Right:

(−1 + 3, 2) = (2, 2)

Left:

(−1 − 3, 2) = (−4, 2)

Top:

(−1, 2 + 3) = (−1, 5)

Bottom:

(−1, 2 − 3) = (−1, −1)

Plot these points and draw a smooth circle through them.

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The Bounding Square

One way to improve a circle sketch is to imagine a square surrounding the circle.

For a circle with radius r:

• the circle extends r units left of the center
• r units right
• r units upward
• r units downward

The resulting square has:

side length = 2r

The circle fits exactly inside this square.

This helps prevent drawing a circle that is too wide, too narrow, or incorrectly centered.

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Finding the Circle's Extreme Coordinates

For a circle centered at:

(h, k)

with radius:

r

the smallest x-coordinate on the circle is:

h − r

The largest x-coordinate is:

h + r

The smallest y-coordinate is:

k − r

The largest y-coordinate is:

k + r

These values are useful when graphing and interpreting circles.

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Worked Example: Extreme Values

Consider:

(x − 4)² + (y + 2)² = 36

Center:

(4, −2)

Radius:

6

Leftmost x-value:

4 − 6 = −2

Rightmost x-value:

4 + 6 = 10

Lowest y-value:

−2 − 6 = −8

Highest y-value:

−2 + 6 = 4

This gives the overall position and size of the circle.

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How the Center Changes the Graph

Compare:

x² + y² = 16

and:

(x − 4)² + y² = 16

Both circles have:

r = 4

Therefore, they have the same size.

The first has center:

(0, 0)

The second has center:

(4, 0)

The second circle has been shifted:

4 units to the right

Changing h or k moves the circle without changing its size.

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Horizontal Shifts

Compare:

x² + y² = 25

with:

(x − 3)² + y² = 25

The second circle moves:

3 units right

Now consider:

(x + 3)² + y² = 25

The center is:

(−3, 0)

so the circle moves:

3 units left

Remember that the sign inside the bracket is opposite the direction of the center coordinate.

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Vertical Shifts

Consider:

x² + (y − 4)² = 9

The center is:

(0, 4)

so the circle has moved:

4 units upward

Now consider:

x² + (y + 4)² = 9

The center is:

(0, −4)

so the circle has moved:

4 units downward

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How the Radius Changes the Graph

Compare:

x² + y² = 4

x² + y² = 16

x² + y² = 36

Their radii are:

√4 = 2

√16 = 4

√36 = 6

Therefore, the circles become larger as r increases.

The center remains:

(0, 0)

Only the size changes.

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Same Center, Different Radii

Circles with the same center but different radii are called:

concentric circles

For example:

x² + y² = 4

x² + y² = 16

x² + y² = 36

all have center:

(0, 0)

but different radii.

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Writing an Equation from a Graph

Suppose a graph shows a circle with:

center = (3, −2)

The circle extends from the center to:

(8, −2)

The radius is the distance between these points:

r = 8 − 3

r = 5

Now use:

(x − h)² + (y − k)² = r²

Substitute:

(x − 3)² + (y + 2)² = 25

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Finding Radius from Two Graph Points

Suppose the center is:

(−2, 4)

and the rightmost point is:

(3, 4)

The horizontal distance is:

3 − (−2) = 5

Therefore:

r = 5

The equation is:

(x + 2)² + (y − 4)² = 25

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Connecting the Equation to Circle Properties

The equation tells us several geometric properties immediately.

For:

(x − 6)² + (y + 3)² = 49

we know:

center = (6, −3)

radius = 7

diameter = 14

leftmost point = (−1, −3)

rightmost point = (13, −3)

top point = (6, 4)

bottom point = (6, −10)

One equation contains a large amount of geometric information.

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Finding the Diameter from an Equation

Consider:

(x + 2)² + (y − 5)² = 100

First:

r² = 100

Therefore:

r = 10

The diameter is:

d = 2r

d = 20

So:

center = (−2, 5)

radius = 10

diameter = 20

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Verifying a Point Using Distance Instead

There are two equivalent ways to test a point.

For a circle centered at:

(2, 1)

with radius:

5

and the point:

(5, 5)

we can substitute into:

(x − 2)² + (y − 1)² = 25

or calculate the distance from the center:

distance² = (5 − 2)² + (5 − 1)²

distance² = 9 + 16

distance² = 25

Therefore:

distance = 5

The point lies exactly one radius from the center, so it lies on the circle.

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Why Squared Distance Is Often Enough

When checking whether a point lies on a circle, you usually do not need to calculate the square root.

For example, if:

r = 7

then:

r² = 49

If the squared distance from the point to the center is also:

49

the point lies on the circle.

This makes calculations faster.

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A Complete Example

Consider:

(x + 3)² + (y − 1)² = 64

Identify the center.

Because:

x + 3 = x − (−3)

the x-coordinate is:

−3

The y-coordinate is:

1

Therefore:

center = (−3, 1)

Now find the radius:

r² = 64

r = 8

Therefore:

radius = 8

and:

diameter = 16

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Graphing the Complete Example

For:

center = (−3, 1)

radius = 8

find the four main points.

Right:

(−3 + 8, 1)

(5, 1)

Left:

(−3 − 8, 1)

(−11, 1)

Top:

(−3, 1 + 8)

(−3, 9)

Bottom:

(−3, 1 − 8)

(−3, −7)

Plot the center and these four points, then draw a smooth circle.

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Complete Point Verification

Using:

(x + 3)² + (y − 1)² = 64

test the point:

(1, 7)

Substitute:

(1 + 3)² + (7 − 1)²

= 4² + 6²

= 16 + 36

= 52

Since:

52 < 64

the point is:

inside the circle

It is not on the circumference.

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Real-World Modeling

Circle equations can model circular objects and regions on coordinate systems.

Examples include:

• circular parks
• fountains
• radar ranges
• Ferris wheels
• circular tracks
• rotating machinery
• restricted zones
• communication ranges
• circular architectural designs

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Real-World Example: Circular Park

Suppose a circular park is represented on a coordinate map.

Its center is:

(20, 30)

and its radius is:

10 units

Its equation is:

(x − 20)² + (y − 30)² = 100

A location can be tested by substituting its coordinates into the equation.

If the left side equals 100, the location is:

on the park boundary

If it is less than 100:

inside the park

If it is greater than 100:

outside the park

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Real-World Example: Communication Range

Suppose a transmitter is located at:

(5, −2)

and has an idealized circular range of:

12 km

The boundary of its range can be modeled by:

(x − 5)² + (y + 2)² = 144

Locations satisfying:

(x − 5)² + (y + 2)² < 144

are inside the idealized coverage region.

The circle equation describes the boundary itself.

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Circle vs Filled Circular Region

This distinction is important.

The equation:

(x − h)² + (y − k)² = r²

describes only the:

circle boundary

The inequality:

(x − h)² + (y − k)² < r²

describes points:

inside the circle

The inequality:

(x − h)² + (y − k)² ≤ r²

describes:

the interior and the boundary

So an equation and an inequality represent different geometric sets.

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A Reliable Strategy for Reading a Circle Equation

Given:

(x − h)² + (y − k)² = r²

use these steps:

1. Check that the equation is in standard form.
2. Identify the x-expression.
3. Reverse its sign to find h.
4. Identify the y-expression.
5. Reverse its sign to find k.
6. Write the center as (h, k).
7. Identify r².
8. Take the positive square root to find r.
9. Check that the radius is positive.
10. Use the center and radius to interpret or graph the circle.

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A Reliable Strategy for Writing a Circle Equation

When given a center and radius:

1. Write the standard form:

   (x − h)² + (y − k)² = r²

2. Identify h and k.
3. Substitute the center coordinates.
4. Be careful with negative coordinates.
5. Substitute the radius.
6. Square the radius.
7. Simplify the signs.
8. Check that the equation gives the correct center and radius.

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A Reliable Strategy for Checking a Point

To determine whether (a, b) lies on a circle:

1. Replace x with a.
2. Replace y with b.
3. Evaluate both squared expressions.
4. Add the results.
5. Compare with r².

If:

result = r² → on

result < r² → inside

result > r² → outside

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A Reliable Strategy for Graphing a Circle

To graph a circle:

1. Identify the center (h, k).
2. Plot the center.
3. Find the radius r.
4. Move r units right from the center.
5. Move r units left.
6. Move r units up.
7. Move r units down.
8. Plot these four points.
9. Sketch a smooth circle through them.
10. Check that the circle is symmetric about horizontal and vertical lines through its center.

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Common Mistakes

Using the signs inside the brackets directly

In:

(x + 4)² + (y − 2)² = 9

the center is:

(−4, 2)

not (4, −2).

Using r² as the radius

If:

r² = 49

then:

r = 7

not 49.

Forgetting to square the radius

If r = 6, the equation must have:

r² = 36

on the right.

Forgetting parentheses with negative coordinates

If h = −3:

x − (−3)

becomes:

x + 3

Thinking x² + y² = 25 has center (25, 25)

The missing h and k values are both zero.

Therefore, the center is:

(0, 0)

Graphing the radius as the diameter

If r = 4, move 4 units from the center, not 8.

Testing only one coordinate

Both x and y coordinates must be substituted.

Taking a negative radius

Radius represents a distance and is positive.

If r² = 36:

r = 6

for the geometric radius.

Assuming every point inside the circle lies on the circle

Only points exactly r units from the center lie on the circumference.

Drawing the circle from appearance alone

Use the center and radius to establish accurate reference points first.

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Key Terms

Circle — set of all points in a plane located the same distance from a fixed center.

Center — fixed point (h, k) from which every point on the circle is the same distance.

Radius — distance from the center to any point on the circle.

Diameter — distance across the circle through its center; equal to 2r.

Standard form of a circle — (x − h)² + (y − k)² = r².

h — x-coordinate of the center.

k — y-coordinate of the center.

r — radius of the circle.

r² — square of the radius and the value appearing on the right side of the standard equation.

Origin — coordinate point (0, 0).

Coordinate plane — two-dimensional system used to locate points using x- and y-coordinates.

Distance formula — relationship used to calculate the distance between two coordinate points.

Concentric circles — circles sharing the same center but having different radii.

Boundary — set of points exactly on the circumference.

Interior — points whose distance from the center is less than the radius.

Exterior — points whose distance from the center is greater than the radius.

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Key Takeaways

• The standard equation of a circle is (x − h)² + (y − k)² = r².
• The center is (h, k).
• The signs inside the brackets appear opposite to the signs of the center coordinates.
• The number on the right side is r², not r.
• Take the positive square root of r² to find the radius.
• A circle centered at the origin has equation x² + y² = r².
• Given the center and radius, substitute h, k, and r into the standard form.
• Negative center coordinates create addition signs inside the corresponding brackets.
• If the diameter is given, divide by 2 before writing the equation.
• If a center and a point on the circle are given, the radius can be found using the distance between them.
• A point lies on a circle when its coordinates make the circle equation true.
• If the squared distance from the center is less than r², the point is inside.
• If the squared distance is greater than r², the point is outside.
• To graph a circle, first plot its center and then locate points one radius away in the four main directions.
• Changing h moves a circle horizontally.
• Changing k moves a circle vertically.
• Changing r changes the size of the circle.
• Circles with the same center but different radii are concentric.
• The circle equation describes the boundary, while an inequality can describe the interior.
• Circle equations connect algebra, coordinate geometry, distance, and real-world spatial modeling.

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Check Your Understanding

1. State the standard form of the equation of a circle and explain what h, k, and r represent.

2. Identify the center and radius of:

   (x − 3)² + (y − 5)² = 49

3. Identify the center and radius of:

   (x + 4)² + (y − 2)² = 81

4. Identify the center and radius of:

   x² + y² = 100

5. Write the equation of a circle with center (2, 6) and radius 5.
6. Write the equation of a circle with center (−3, 4) and radius 7.
7. Write the equation of a circle centered at (−2, −5) with diameter 12.
8. A circle has center (1, 2) and passes through (4, 6). Determine its radius and write its equation.
9. Determine whether (3, 4) lies on the circle x² + y² = 25.
10. Determine whether (2, 1) is inside, on, or outside the circle x² + y² = 16.
11. For (x − 5)² + (y + 2)² = 36, determine the center, radius, diameter, leftmost point, rightmost point, highest point, and lowest point.
12. Describe how the graph of x² + y² = 25 changes when the equation becomes (x − 4)² + y² = 25.
13. Describe how the graph of (x − 2)² + (y + 3)² = 16 changes if the right side becomes 64.
14. A student says that (x + 6)² + (y − 1)² = 25 has center (6, −1) and radius 25. Identify both errors and give the correct center and radius.
15. A circular safety zone has center (10, −4) and radius 8 km. Write an equation for its boundary and explain how you could determine whether a location at (14, 2) lies inside, on, or outside the safety zone.