Finding Centers and Radii

The equation of a circle contains two of its most important geometric properties:

• its center, which determines its location
• its radius, which determines its size

When a circle is written in standard form, both pieces of information can usually be identified directly.

The standard form is:

(x − h)² + (y − k)² = r²

where:

• (h, k) = center
• r = radius

Learning to read this equation quickly allows us to graph, construct, compare, and analyze circles.

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Identifying the Center

In:

(x − h)² + (y − k)² = r²

the center is:

(h, k)

For example:

(x − 4)² + (y − 3)² = 25

gives:

h = 4

k = 3

Therefore:

center = (4, 3)

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Watch the Signs

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

For example:

(x + 5)² + (y − 2)² = 16

can be thought of as:

[x − (−5)]² + (y − 2)² = 16

Therefore:

center = (−5, 2)

This sign reversal is one of the most important ideas when reading circle equations.

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Reading Different Sign Combinations

Consider the following examples:

The number inside each bracket tells you the coordinate after you reverse its sign.

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Finding the Radius

The right side of the standard equation is:

r²

Therefore, to find the radius, take the positive square root.

For example:

(x − 2)² + (y + 1)² = 49

gives:

r² = 49

r = √49

r = 7

The radius is not 49.

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Why the Radius Is Positive

Solving:

r² = 49

algebraically could produce:

r = ±7

However, the geometric radius represents a:

distance

A distance cannot be negative.

Therefore, the radius of the circle is:

r = 7

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Worked Example: Center and Radius

Find the center and radius of:

(x − 6)² + (y + 2)² = 81

Identify the center:

x − 6 → h = 6

y + 2 → k = −2

Therefore:

center = (6, −2)

Now find the radius:

r² = 81

r = √81

radius = 9

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Worked Example with Two Negative Coordinates

Find the center and radius of:

(x + 4)² + (y + 7)² = 36

The center is:

(−4, −7)

because:

x + 4 = x − (−4)

y + 7 = y − (−7)

The radius is:

r = √36

r = 6

Therefore:

center = (−4, −7)

radius = 6

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Worked Example with a Non-Perfect Square

Consider:

(x − 1)² + (y − 5)² = 40

Center:

(1, 5)

Radius:

r = √40

Simplify:

r = √(4 × 10)

r = 2√10

Approximately:

r ≈ 6.32

Not every circle has a whole-number radius.

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Circles Centered at the Origin

If the center is:

(0, 0)

then:

h = 0

k = 0

and the equation simplifies to:

x² + y² = r²

For example:

x² + y² = 64

has:

center = (0, 0)

radius = 8

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When Only One Coordinate Is Zero

Consider:

x² + (y − 5)² = 16

The x-part can be written:

(x − 0)²

Therefore:

h = 0

k = 5

The center is:

(0, 5)

and:

r = 4

Now consider:

(x + 3)² + y² = 100

This gives:

center = (−3, 0)

radius = 10

A missing number inside a squared variable term means the corresponding center coordinate is zero.

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Connecting the Equation to the Graph

Once the center and radius are known, the circle can be located on a coordinate plane.

For:

(x − 3)² + (y + 2)² = 25

we have:

center = (3, −2)

radius = 5

From the center, the circle extends 5 units:

• left
• right
• up
• down

This provides four useful points for constructing the circle.

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Finding the Four Main Points

For a circle with 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)

These points help determine the circle's location and size.

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Worked Example: Constructing a Circle

Consider:

(x − 2)² + (y − 4)² = 9

Center:

(2, 4)

Radius:

3

Find the four main points.

Right:

(2 + 3, 4)

(5, 4)

Left:

(2 − 3, 4)

(−1, 4)

Top:

(2, 4 + 3)

(2, 7)

Bottom:

(2, 4 − 3)

(2, 1)

Plot these four points around the center and draw a smooth circle through them.

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Constructing a Circle from Coordinate Information

Suppose you are told:

center = (−3, 2)

and:

radius = 4

You can construct the circle by first plotting:

(−3, 2)

Then move 4 units in each main direction.

Right:

(1, 2)

Left:

(−7, 2)

Top:

(−3, 6)

Bottom:

(−3, −2)

The equation is:

(x + 3)² + (y − 2)² = 16

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Constructing a Circle from Center and Diameter

Suppose a circle has:

center = (5, −1)

diameter = 12

First calculate the radius:

r = d/2

r = 12/2

r = 6

Therefore:

r² = 36

The equation is:

(x − 5)² + (y + 1)² = 36

This circle extends 6 coordinate units in every direction from its center.

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Constructing a Circle from Center and One Point

Suppose the center is:

(2, 3)

and the circle passes through:

(6, 6)

The radius is the distance from the center to the point.

Instead of immediately finding r, we can first find r²:

r² = (6 − 2)² + (6 − 3)²

r² = 4² + 3²

r² = 16 + 9

r² = 25

Therefore:

r = 5

The circle equation is:

(x − 2)² + (y − 3)² = 25

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Why Coordinate Information Is Enough

A circle is completely determined if we know:

• its center
• its radius

Once those two properties are known, there is only one possible circle.

Similarly, if we know the center and one point on the circle, the distance between those points determines the radius.

This is why coordinate information can be used to construct an exact circle.

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Finding the Diameter from an Equation

Consider:

(x + 2)² + (y − 8)² = 121

Center:

(−2, 8)

Radius:

r = √121

r = 11

Diameter:

d = 2r

d = 22

Therefore:

diameter = 22

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Finding the Circle's Horizontal Range

Consider:

(x − 4)² + (y − 1)² = 36

Center:

(4, 1)

Radius:

6

The circle extends from:

h − r

to:

h + r

Therefore:

minimum x = 4 − 6 = −2

maximum x = 4 + 6 = 10

So the circle extends horizontally from:

x = −2 to x = 10

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Finding the Circle's Vertical Range

Using the same circle:

center = (4, 1)

radius = 6

The minimum y-coordinate is:

k − r

1 − 6 = −5

The maximum y-coordinate is:

k + r

1 + 6 = 7

Therefore, the circle extends vertically from:

y = −5 to y = 7

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Finding Extreme Points

For:

(x − 4)² + (y − 1)² = 36

the extreme points are:

Left:

(−2, 1)

Right:

(10, 1)

Top:

(4, 7)

Bottom:

(4, −5)

These points are useful when solving problems involving the position of a circle.

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Comparing Circle Locations

Consider:

Circle A:

(x − 2)² + (y − 3)² = 16

Circle B:

(x − 7)² + (y − 3)² = 16

Circle A has center:

(2, 3)

Circle B has center:

(7, 3)

Both have:

r = 4

Therefore:

• they have the same size
• Circle B is 5 units to the right of Circle A

Only the center has changed.

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Comparing Circle Sizes

Consider:

Circle A:

(x − 2)² + (y − 3)² = 16

Circle B:

(x − 2)² + (y − 3)² = 49

Both have center:

(2, 3)

Circle A:

r = 4

Circle B:

r = 7

Therefore:

Circle B is larger

The circles are concentric because they share the same center.

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Comparing Both Location and Size

Consider:

Circle A:

(x + 1)² + (y − 4)² = 25

Circle B:

(x − 5)² + (y + 2)² = 64

Circle A:

center = (−1, 4)

radius = 5

Circle B:

center = (5, −2)

radius = 8

Therefore:

• the centers are different
• Circle B is larger
• Circle B is positioned farther right and lower than Circle A

Comparing circle equations lets us compare both:

location and size

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Comparing Circles Efficiently

When comparing two circle equations, create a simple summary.

Then ask:

• Do they have the same center?
• Do they have the same radius?
• Which is larger?
• How are their centers positioned relative to each other?

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Congruent Circles

Two circles are congruent if they have the same radius.

They do not need to have the same center.

For example:

(x − 2)² + y² = 25

and:

(x + 7)² + (y − 4)² = 25

both have:

r = 5

Therefore, the circles are:

congruent

They have the same size but different locations.

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Concentric Circles

Two or more circles are concentric if they have the same center but different radii.

For example:

(x − 3)² + (y + 1)² = 9

and:

(x − 3)² + (y + 1)² = 64

both have center:

(3, −1)

Their radii are:

3 and 8

Therefore, they are:

concentric circles

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Same Center and Same Radius

Suppose:

Circle A:

(x − 4)² + (y + 2)² = 36

Circle B:

(x − 4)² + (y + 2)² = 36

Both have:

center = (4, −2)

radius = 6

They are not two different circles.

They are:

the same circle

A circle is uniquely determined by its center and radius.

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Distance Between Circle Centers

Sometimes comparing circle locations requires finding the distance between their centers.

Suppose:

Circle A has center:

(1, 2)

Circle B has center:

(5, 5)

The horizontal change is:

5 − 1 = 4

The vertical change is:

5 − 2 = 3

Using the distance formula:

distance = √(4² + 3²)

distance = √25

distance = 5

The centers are 5 units apart.

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Comparing Distance with Radii

Knowing the distance between centers can help us reason about whether circles are separate, touching, or overlapping.

Suppose two circles have:

r₁ = 3

r₂ = 2

and the centers are:

5 units apart.

Then:

r₁ + r₂ = 3 + 2

r₁ + r₂ = 5

Since the center distance equals the sum of the radii, the circles touch externally at:

exactly one point

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Separate Circles

If the distance between centers is greater than the sum of the radii:

d > r₁ + r₂

the circles are separate.

For example:

r₁ = 3

r₂ = 4

center distance = 10

Since:

10 > 3 + 4

10 > 7

the circles do not touch or overlap.

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Intersecting Circles

If:

|r₁ − r₂| < d < r₁ + r₂

the circles intersect at two points.

For example:

r₁ = 5

r₂ = 4

d = 6

Difference:

|5 − 4| = 1

Sum:

5 + 4 = 9

Since:

1 < 6 < 9

the circles intersect at:

two points

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External Tangency

Two circles are externally tangent when they touch at exactly one point from the outside.

This occurs when:

d = r₁ + r₂

Example:

r₁ = 5

r₂ = 3

d = 8

Since:

5 + 3 = 8

the circles are externally tangent.

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Internal Tangency

One circle may lie inside another and touch it at exactly one point.

This occurs when:

d = |r₁ − r₂|

For example:

r₁ = 8

r₂ = 3

d = 5

Since:

|8 − 3| = 5

the circles are:

internally tangent

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One Circle Inside Another

If:

d < |r₁ − r₂|

the smaller circle lies completely inside the larger circle without touching it.

For example:

r₁ = 8

r₂ = 3

d = 2

Since:

2 < |8 − 3|

2 < 5

the smaller circle lies completely inside the larger circle.

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Circle Position Summary

Let:

d = distance between centers

Then:

This provides a powerful way to solve problems involving the location and size of circles.

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Worked Comparison Problem

Consider:

Circle A:

x² + y² = 16

Circle B:

(x − 7)² + y² = 9

Circle A:

center = (0, 0)

radius = 4

Circle B:

center = (7, 0)

radius = 3

Distance between centers:

d = 7

Sum of radii:

r₁ + r₂ = 4 + 3 = 7

Therefore:

d = r₁ + r₂

The circles are:

externally tangent

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Worked Overlap Problem

Circle A:

(x − 1)² + (y − 2)² = 25

Circle B:

(x − 5)² + (y − 2)² = 16

Centers:

A = (1, 2)

B = (5, 2)

Distance between centers:

d = 4

Radii:

r₁ = 5

r₂ = 4

Calculate:

|5 − 4| = 1

5 + 4 = 9

Since:

1 < 4 < 9

the circles:

intersect at two points

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Real-World Example: Communication Zones

Suppose two wireless transmitters are modeled using circular coverage areas.

Transmitter A:

center = (2, 4)

radius = 6 km

Transmitter B:

center = (10, 4)

radius = 5 km

The centers are:

8 km apart.

The sum of the radii is:

6 + 5 = 11 km

The difference is:

|6 − 5| = 1 km

Since:

1 < 8 < 11

the two circular coverage boundaries intersect.

Their coverage regions also overlap.

This type of model can help analyze service areas.

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Real-World Example: Circular Safety Zones

Suppose a safety zone has equation:

(x − 20)² + (y − 15)² = 100

The center is:

(20, 15)

The radius is:

10 units

Therefore, the zone's boundary is 10 units from the central location in every direction.

Its diameter is:

20 units

A point can be tested to determine whether it lies inside, on, or outside the safety boundary.

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Testing a Location

Using:

(x − 20)² + (y − 15)² = 100

consider the location:

(26, 23)

Calculate:

(26 − 20)² + (23 − 15)²

= 6² + 8²

= 36 + 64

= 100

Therefore:

the point lies exactly on the circle

It is exactly 10 units from the center.

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Inside or Outside a Circle

For:

(x − h)² + (y − k)² = r²

a point is:

inside if:

(x − h)² + (y − k)² < r²

on if:

(x − h)² + (y − k)² = r²

outside if:

(x − h)² + (y − k)² > r²

This allows us to solve many location problems without graphing.

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Solving a Boundary Problem

A circular park has center:

(5, 6)

and radius:

8 m

A lamp is located at:

(9, 9)

Determine whether the lamp is inside the park boundary.

Calculate its squared distance from the center:

(9 − 5)² + (9 − 6)²

= 4² + 3²

= 16 + 9

= 25

Compare with:

r² = 8² = 64

Since:

25 < 64

the lamp is:

inside the circular park

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Solving a Location Problem Without Square Roots

Notice that we did not calculate:

√25

We compared:

distance² = 25

with:

radius² = 64

Because:

25 < 64

we immediately know that the point is inside.

Comparing squared distances often makes circle problems faster and simpler.

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Translating a Circle

Suppose:

x² + y² = 25

has center:

(0, 0)

and radius:

5.  

Now consider:

(x − 4)² + (y + 3)² = 25

The new center is:

(4, −3)

The radius remains:

5

Therefore, the circle has been moved:

• 4 units right
• 3 units down

without changing its size.

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Changing Only the Radius

Compare:

(x − 2)² + (y − 1)² = 9

and:

(x − 2)² + (y − 1)² = 49

Both centers are:

(2, 1)

The first radius is:

3

The second radius is:

7

Therefore, the second circle is larger but has not moved.

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Changing Only the Center

Compare:

(x − 2)² + (y − 1)² = 25

and:

(x + 4)² + (y − 6)² = 25

Both radii are:

5

Therefore, the circles are congruent.

However, their centers are:

(2, 1)

and:

(−4, 6)

so their locations are different.

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A Reliable Strategy for Finding Center and Radius

When given a standard circle equation:

1. Compare it with (x − h)² + (y − k)² = r².
2. Identify the x-expression.
3. Reverse the sign to find h.
4. Identify the y-expression.
5. Reverse the sign to find k.
6. Write the center as (h, k).
7. Identify r².
8. Take the positive square root.
9. State the radius.
10. Check whether the result makes sense.

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A Reliable Strategy for Constructing a Circle

When given coordinate information:

1. Identify or calculate the center.
2. Identify or calculate the radius.
3. Plot the center.
4. Move r units right.
5. Move r units left.
6. Move r units upward.
7. Move r units downward.
8. Plot the four extreme points.
9. Draw a smooth circle.
10. Write the equation if required.

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A Reliable Strategy for Comparing Circles

For each circle:

1. identify the center
2. identify the radius
3. compare the center coordinates
4. compare the radii
5. determine whether they are the same size
6. determine how their locations differ
7. calculate the distance between centers if necessary
8. compare that distance with the radii
9. determine whether the circles are separate, tangent, intersecting, or nested

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Common Mistakes

Reading the signs directly

For:

(x + 4)² + (y − 3)² = 25

the center is:

(−4, 3)

not (4, −3).

Using r² as the radius

If:

r² = 64

then:

r = 8

not 64.

Using a negative radius

Radius is a distance and must be positive.

Forgetting that x² means (x − 0)²

If no horizontal shift appears, the x-coordinate of the center is zero.

Forgetting that y² means (y − 0)²

If no vertical shift appears, the y-coordinate of the center is zero.

Using the diameter as the radius

If d = 14:

r = 7

Moving the circle when only the radius changes

Changing r changes size, not center.

Changing the size when only the center changes

If r² remains unchanged, the circle remains the same size.

Comparing the right-side numbers instead of the radii

For circle size, compare:

√r²

although larger positive r² also corresponds to a larger radius.

Judging circle relationships only from a sketch

Coordinate calculations are more reliable than appearance.

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Key Terms

Center — fixed point (h, k) that determines the location of a circle.

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

Diameter — twice the radius.

Standard form — circle equation written as (x − h)² + (y − k)² = r².

Origin — point (0, 0).

Coordinate plane — system for locating points using x- and y-coordinates.

Extreme points — leftmost, rightmost, highest, and lowest points of a circle.

Congruent circles — circles with equal radii.

Concentric circles — circles sharing the same center.

Distance between centers — straight-line distance connecting the centers of two circles.

External tangency — two circles touching externally at exactly one point.

Internal tangency — one circle inside another while touching at exactly one point.

Intersecting circles — circles whose circumferences share two points.

Squared distance — distance calculation before taking the square root; useful for comparing a point with r².

Translation — movement of a figure without changing its size or shape.

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Key Takeaways

• A circle in standard form is written as (x − h)² + (y − k)² = r².
• The center is (h, k).
• Signs inside the brackets are opposite the corresponding center coordinates.
• The right side represents r², not r.
• Radius is found by taking the positive square root of the right side.
• A circle centered at the origin has equation x² + y² = r².
• Missing horizontal or vertical shifts correspond to center coordinates of zero.
• A circle is completely determined by its center and radius.
• If the center and one point on the circle are known, the radius can be calculated using distance.
• The radius determines the size of a circle.
• The center determines its location.
• Circles with equal radii are congruent.
• Circles with the same center are concentric if their radii differ.
• Circles with the same center and same radius are the same circle.
• The four extreme points can be found by adding and subtracting the radius from the center coordinates.
• Comparing the distance between centers with the radii can determine whether circles are separate, tangent, intersecting, or nested.
• A point's squared distance from the center can determine whether it lies inside, on, or outside a circle.
• Circle equations can model real-world boundaries, ranges, zones, and circular structures.
• Algebraic information in a circle equation provides precise information about both location and size.

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Check Your Understanding

1. Identify the center and radius of (x − 4)² + (y − 7)² = 36.
2. Identify the center and radius of (x + 3)² + (y − 5)² = 81.
3. Identify the center and radius of (x + 6)² + (y + 2)² = 49.
4. Determine the center, radius, and diameter of x² + y² = 144.
5. Determine the center and radius of x² + (y − 8)² = 25.
6. Write the equation of a circle with center (−4, 3) and radius 6.
7. A circle has center (5, −2) and diameter 18. Write its equation.
8. A circle has center (1, 2) and passes through (4, 6). Determine its radius and equation.
9. For (x − 3)² + (y + 4)² = 16, determine the leftmost, rightmost, highest, and lowest points.
10. Compare (x − 2)² + (y − 1)² = 25 and (x + 5)² + (y − 1)² = 25. How are the circles alike, and how are they different?
11. Compare (x − 4)² + (y + 2)² = 9 and (x − 4)² + (y + 2)² = 64. Describe their centers, radii, and relationship.
12. Two circles have radii 5 and 3, and their centers are 8 units apart. Describe the relationship between the circles and justify your answer.
13. Two circles have radii 7 and 4, and their centers are 5 units apart. Determine whether they are separate, tangent, intersecting, or nested.
14. A circular safety boundary has center (10, 6) and radius 8 units. Determine whether the point (16, 10) is inside, on, or outside the boundary.
15. A student says that (x + 5)² + (y − 2)² = 100 has center (5, −2) and radius 100. Identify both errors, give the correct center and radius, and explain how the equation determines the circle's location and size.