Properties of Circles

A circle is a geometric figure made from all points in a plane that are the same distance from one fixed point.

That fixed point is called the center.

The constant distance from the center to the circle is called the radius.

Circles appear throughout mathematics and in everyday objects such as wheels, clocks, gears, coins, pipes, round tables, lenses, and circular tracks.

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Defining a Circle Geometrically

A circle can be defined precisely as:

the set of all points in a plane that are the same distance from a fixed point

The fixed point is the:

center

The fixed distance is the:

radius

For example, if O is the center and every point on the circle is 5 cm from O, then the circle has:

radius = 5 cm

Every point on the circumference must be exactly 5 cm from the center.

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Circle vs Circular Region

In geometry, the word circle technically refers to the boundary itself.

The region inside the circle is sometimes called the:

disk

In everyday language, however, people often use "circle" to refer to the entire shape, including its interior.

This distinction becomes useful when studying more advanced geometry.

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The Center

The center is the fixed point that is equally distant from every point on the circle.

It is commonly labeled with a capital letter such as:

O

If A, B, and C are points on a circle with center O, then:

OA = OB = OC

because each of these distances is a radius.

The center itself is not part of the circumference.

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The Radius

A radius is a line segment from the center of a circle to any point on the circle.

If O is the center and A lies on the circle, then:

OA is a radius

A circle contains infinitely many possible radii.

All radii of the same circle have:

equal length

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Radius as a Length and a Segment

The word radius can refer to:

• the segment from the center to the circle
• the length of that segment

For example:

"The radius is 6 cm."

means the distance from the center to any point on the circle is:

6 cm

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The Diameter

A diameter is a line segment that:

• has both endpoints on the circle
• passes through the center

Suppose A and B are points on a circle and segment AB passes through center O.

Then:

AB is a diameter

A diameter can be thought of as two radii placed end to end.

Therefore:

diameter = 2 × radius

Using symbols:

d = 2r

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Radius and Diameter

The relationship between radius and diameter is fundamental.

d = 2r

Therefore:

r = d/2

This means:

• the diameter is twice the radius
• the radius is half the diameter

For example, if:

r = 7 cm

then:

d = 2(7)

d = 14 cm

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Worked Example: Finding Diameter

A circular plate has a radius of:

12 cm

Find its diameter.

Use:

d = 2r

Substitute:

d = 2(12)

d = 24 cm

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Worked Example: Finding Radius

A bicycle wheel has a diameter of:

70 cm

Find its radius.

Use:

r = d/2

Substitute:

r = 70/2

r = 35 cm

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Worked Example with Decimals

A circular pipe has an inside diameter of:

8.6 cm

Find its radius.

r = d/2

r = 8.6/2

r = 4.3 cm

The radius and diameter must use the same unit.

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The Diameter Is a Special Chord

A chord is a line segment whose two endpoints lie on the circle.

A diameter also has two endpoints on the circle.

Therefore:

every diameter is a chord

However:

not every chord is a diameter

A chord is only a diameter if it passes through the center.

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Chords

A circle contains many possible chords.

Some chords pass near the center.

Others lie closer to the circumference.

The diameter is the chord that passes through the center.

It is also:

the longest possible chord in a circle

As a chord moves closer to the center, its length increases.

A chord passing exactly through the center becomes a diameter.

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Why the Diameter Is the Longest Chord

A diameter stretches from one side of the circle to the opposite side while passing directly through the center.

Its length is:

2r

A chord that does not pass through the center takes a shorter path across the circle.

Therefore:

diameter = longest chord

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The Tangent

A tangent is a line that touches a circle at exactly one point.

That point is called the:

point of tangency

For example, if a line touches a circle only at point T, then:

T = point of tangency

and the line is:

tangent to the circle

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Tangent and Radius Relationship

An important property of circles is:

a tangent is perpendicular to the radius drawn to the point of tangency

If O is the center and T is the point of tangency:

OT ⟂ tangent line

Therefore, the angle between the radius and tangent is:

90°

This relationship is extremely useful in circle geometry.

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Real-World Tangent Example

Imagine a bicycle wheel touching a perfectly flat road.

The wheel and road touch at one point.

At that instant, the road can be modeled as a:

tangent to the circular wheel

The radius drawn from the wheel's center to the contact point is perpendicular to the road.

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The Secant

A secant is a line that passes through a circle and intersects it at:

two points

This is different from a tangent.

A tangent intersects the circle once.

A secant intersects the circle twice.

The secant continues beyond the circle in both directions.

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Tangent vs Secant

A secant that passes through the center contains a diameter as part of the line.

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Chord vs Secant

Chords and secants are related, but they are not the same.

A chord is a:

line segment

with both endpoints on the circle.

A secant is an entire:

line

that crosses the circle at two points.

The part of a secant between its two intersection points is:

a chord

This relationship is important when identifying parts of circle diagrams.

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Segment vs Line

Geometry distinguishes carefully between a line and a line segment.

A line:

• extends forever in both directions

A line segment:

• has two endpoints

Therefore:

radius → segment

diameter → segment

chord → segment

tangent → line

secant → line

In diagrams, arrows at the ends often indicate that something is a line.

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The Circumference

The circumference is the distance around a circle.

It is similar to the perimeter of other two-dimensional shapes.

Although circumference calculations are often studied as a separate topic, the radius and diameter are directly connected to circumference.

The common formulas are:

C = 2πr

or:

C = πd

This makes sense because:

d = 2r

so the two formulas are equivalent.

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Understanding π

The symbol:

π

is read as:

pi

For every circle:

circumference/diameter = π

Approximately:

π ≈ 3.14159

This ratio is the same for every circle, regardless of its size.

A coin and a giant circular stadium both have the same circumference-to-diameter ratio.

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Parts of a Circle Compared

These definitions help distinguish features that can look similar in diagrams.

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Identifying a Radius in a Diagram

To determine whether a segment is a radius, ask:

Does one endpoint lie at the center?

and:

Does the other endpoint lie on the circle?

If both are true:

the segment is a radius

A segment connecting two points on the circle is not normally called a radius unless one endpoint is also the center, which is not on the circle.

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Identifying a Diameter in a Diagram

Ask:

1. Are both endpoints on the circle?
2. Does the segment pass through the center?

If both are true:

it is a diameter

If it connects two points on the circle but does not pass through the center:

it is a chord

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Identifying a Chord in a Diagram

Ask:

Are both endpoints on the circle?

If yes, the segment is a chord.

Then ask:

Does it pass through the center?

If yes, it is also a diameter.

Therefore, a single segment can sometimes have more than one correct classification.

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Identifying Tangents and Secants

Count the number of points where the line meets the circle.

If the line touches the circle at exactly:

one point → tangent

If it crosses the circle at:

two points → secant

This is one of the quickest ways to distinguish them.

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Important Relationships Among Circle Parts

Several circle properties are connected.

Radius and diameter

d = 2r

Diameter and chord

Every diameter is a chord.

Longest chord

A diameter is the longest chord.

Tangent and radius

A tangent is perpendicular to the radius at the point of tangency.

Secant and chord

The portion of a secant inside the circle is a chord.

These relationships allow us to reason about diagrams rather than simply memorize vocabulary.

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Equal Radii

Suppose O is the center of a circle and A, B, C, and D lie on the circle.

Then:

OA = OB = OC = OD

because all four segments are radii of the same circle.

If:

OA = 9 cm

then:

OB = 9 cm

OC = 9 cm

OD = 9 cm

and:

diameter = 18 cm

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Using Radius to Find Missing Lengths

Suppose O is the center and AB is a diameter.

If:

OA = 6.5 cm

then:

OB = 6.5 cm

because both are radii.

Therefore:

AB = OA + OB

AB = 6.5 + 6.5

AB = 13 cm

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Using Diameter to Find Missing Lengths

Suppose CD is a diameter measuring:

22 cm

The center O lies halfway along the diameter.

Therefore:

CO = OD

and:

CO = 22/2

CO = 11 cm

Each half of a diameter is a radius.

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The Center Is the Midpoint of Every Diameter

Every diameter passes through the center.

Because the two halves of a diameter are radii of equal length, the center divides every diameter into two equal segments.

Therefore:

the center is the midpoint of every diameter

If AB is a diameter with center O:

AO = OB

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Circles of Different Sizes

The size of a circle is determined by its radius.

If two circles have equal radii:

they are the same size

If one circle has a larger radius:

it is a larger circle

Changing the center moves the circle.

Changing the radius changes its size.

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Concentric Circles

Concentric circles are circles that share the same center but have different radii.

Examples can appear in:

• archery targets
• dartboards
• ripples
• circular designs
• mechanical components

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Circles in Coordinate Geometry

A circle can also be placed on a coordinate plane.

If its center is:

(h, k)

and its radius is:

r

then every point (x, y) on the circle is exactly r units from the center.

This leads to the standard circle equation:

(x − h)² + (y − k)² = r²

This equation comes from the Pythagorean theorem.

For this topic, the important idea is that the equation represents the same geometric definition:

every point on the circle is the same distance from the center.

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Example of a Circle on a Coordinate Plane

Consider:

(x − 2)² + (y + 1)² = 25

The center is:

(2, −1)

because:

y + 1 = y − (−1)

The radius satisfies:

r² = 25

Therefore:

r = 5

The diameter is:

d = 10

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Real-World Circles

Circles appear in many everyday objects.

Examples include:

• bicycle wheels
• car wheels
• clocks
• coins
• circular tables
• plates
• gears
• pipes
• lenses
• roundabouts
• ferris wheels
• CDs and discs
• circular tracks

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Wheels

A wheel provides an excellent example of circle terminology.

The axle is located near the:

center

A spoke can represent a:

radius

A straight segment extending from one side of the wheel through the center to the opposite side represents a:

diameter

The road touching the wheel can be modeled as a:

tangent

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Clocks

A circular clock also illustrates circle properties.

The point where the hands rotate is the:

center

The distance from the center to the outer edge is the:

radius

A straight line across the clock face through its center is a:

diameter

A straight segment connecting two points on the edge without necessarily passing through the center is a:

chord

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Pipes and Cylinders

The circular end of a pipe can be analyzed using circle geometry.

The distance across the pipe through its center is the:

diameter

Half this distance is the:

radius

This is important in:

• construction
• plumbing
• manufacturing
• engineering

For example, a pipe with diameter 16 cm has:

r = 16/2

r = 8 cm

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Tangents in Real-World Design

Tangents appear wherever a straight path meets a circular shape smoothly.

Examples include:

• a road touching a roundabout
• a belt touching a circular pulley
• a wheel touching a road
• mechanical designs involving gears or rollers

The radius to the contact point is perpendicular to the tangent.

This property allows engineers to determine directions and angles accurately.

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Secants in Real-World Models

A straight path passing through a circular region can be modeled as a secant.

For example:

• a road passing through a circular park
• a laser beam passing through a circular lens cross-section
• a straight survey line crossing a circular boundary

The two points where the path crosses the boundary are the secant's intersection points.

The portion between them forms a chord.

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Comparing Radius, Diameter, and Chord

Suppose a circle has radius:

8 cm

Then every radius has length:

8 cm

Every diameter has length:

16 cm

Other chords can have different lengths, but no chord can be longer than:

16 cm

Therefore:

diameter = longest chord = 16 cm

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Reasoning from a Diagram

Suppose O is the center.

A and B lie on the circle.

AB passes through O.

Then:

OA = radius

OB = radius

AB = diameter

AB is also a chord.

If another segment CD joins two points on the circle but does not pass through O:

CD = chord

but:

CD is not a diameter.

This demonstrates that geometric objects can belong to more than one category.

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A Complete Circle Example

A circular garden has a diameter of:

18 m

Its radius is:

r = 18/2

r = 9 m

A straight path from one edge of the garden to another through the center is a:

diameter

A straight path joining two boundary points without passing through the center is a:

chord

A straight road touching the garden at exactly one point is a:

tangent

A straight road passing through the garden and crossing its boundary twice is a:

secant

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Recognizing Information from Diagrams

When looking at a circle diagram, do not rely only on appearance.

Look for mathematical information such as:

• center labels
• endpoints
• right-angle symbols
• equal-length marks
• arrows showing lines
• points of intersection

For example, a line may look as though it touches the circle, but if the diagram does not clearly show or state tangency, you should not automatically assume it is tangent.

Geometry diagrams are often not drawn perfectly to scale.

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A Strategy for Identifying Circle Parts

When examining a circle diagram:

1. Locate the center.
2. Identify segments from the center to the circumference.
3. Label these as radii.
4. Look for segments joining two points on the circle.
5. Determine whether any pass through the center.
6. Classify those as diameters.
7. Classify other such segments as chords.
8. Look for lines meeting the circle once.
9. Identify these as tangents.
10. Look for lines meeting the circle twice.
11. Identify these as secants.
12. Use markings and labels rather than judging only by appearance.

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A Strategy for Radius and Diameter Problems

If the radius is known:

multiply by 2

d = 2r

If the diameter is known:

divide by 2

r = d/2

Then:

1. write the appropriate relationship
2. substitute the known value
3. calculate
4. include the correct unit
5. check whether the answer is reasonable

The diameter should always be twice the radius of the same circle.

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Common Mistakes

Thinking the center is on the circle

The center lies inside the circle and is equally distant from every point on it.

Confusing radius and diameter

The diameter is twice the radius.

Thinking every chord is a diameter

Every diameter is a chord, but only chords passing through the center are diameters.

Thinking a chord must pass through the center

A chord only needs its endpoints on the circle.

Calling a secant a chord

A secant is an entire line. The portion inside the circle is a chord.

Thinking a tangent passes slightly through the circle

A tangent meets the circle at exactly one point.

Forgetting the tangent-radius relationship

The radius to the point of tangency is perpendicular to the tangent.

Thinking a secant intersects once

A secant intersects the circle at two points.

Forgetting that all radii of the same circle are equal

Every point on a circle is the same distance from its center.

Assuming a diagram is perfectly drawn to scale

Use labels, measurements, and mathematical markings rather than appearance alone.

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Key Terms

Circle — set of all points in a plane that are the same distance from a fixed point.

Center — fixed point equally distant from every point on a circle.

Radius — segment from the center to a point on the circle, or its length.

Diameter — chord passing through the center of a circle.

Chord — segment with both endpoints on the circle.

Tangent — line that touches a circle at exactly one point.

Point of tangency — single point where a tangent meets a circle.

Secant — line that intersects a circle at two points.

Circumference — distance around a circle.

Disk — region inside and including a circle.

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

Perpendicular — meeting at a 90° angle.

Midpoint — point dividing a segment into two equal parts.

π (pi) — constant ratio of a circle's circumference to its diameter, approximately 3.14159.

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Key Takeaways

• A circle consists of all points in a plane that are the same distance from its center.
• The radius is the fixed distance from the center to the circle.
• All radii of the same circle have equal lengths.
• The diameter passes through the center and connects two points on the circle.
• The diameter is twice the radius: d = 2r.
• The radius is half the diameter: r = d/2.
• The center is the midpoint of every diameter.
• A chord joins two points on a circle.
• Every diameter is a chord.
• Not every chord is a diameter.
• The diameter is the longest possible chord.
• A tangent touches a circle at exactly one point.
• A tangent is perpendicular to the radius at the point of tangency.
• A secant intersects a circle at two points.
• The part of a secant inside the circle forms a chord.
• Horizontal, vertical, or diagonal orientation does not change whether a segment is a radius, diameter, or chord.
• Circle properties can be identified using definitions, labels, intersection points, and geometric markings.
• Circles and their properties appear in wheels, clocks, pipes, gears, roads, architecture, and engineering.
• The geometric definition of a circle also forms the basis for circle equations in coordinate geometry.

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Check Your Understanding

1. Define a circle geometrically.
2. Explain the difference between the center and the radius of a circle.
3. A circle has radius 9 cm. Calculate its diameter.
4. A circular wheel has diameter 64 cm. Calculate its radius.
5. Explain why every diameter is a chord but not every chord is a diameter.
6. What is the longest possible chord in a circle? Explain why.
7. Describe the difference between a chord and a secant.
8. Describe the difference between a tangent and a secant.
9. A tangent touches a circle at point T. The center is O. What angle is formed between OT and the tangent?
10. A circle has center O and radius 7.5 cm. Points A, B, and C lie on the circle. Determine OA, OB, and OC.
11. A diameter AB passes through center O and AB = 26 cm. Determine AO and OB.
12. A straight line intersects a circle at points P and Q and continues beyond both points. What type of line is it? What is segment PQ?
13. A bicycle wheel has radius 35 cm. Identify what parts of the wheel could represent the center, radius, diameter, and a tangent when the wheel rests on a flat road.
14. A student says, "A chord and a diameter are completely different because a chord never passes through the center." Explain the error.
15. A circle has diameter 18 m. A straight path crosses from one side of the circle to the other through the center, while another straight road touches the circle at only one point. Identify each geometric feature and explain the properties that justify your answer.