Basic Concepts of a Circle

Take a pencil compass. Keep its pointed end fixed at a point on your paper and rotate the pencil end once completely around that point, keeping the compass opening unchanged. The pencil traces the familiar closed curve called a circle.

Notice what happens: the pointed end stays fixed, and the pencil tip remains at the same distance from it throughout. This gives us the basic idea of a circle.

Pencil compass

Drawing a circle with a pencil compass

A circle is the set of all points in a plane at a fixed positive distance from a fixed point.

⭐ Centre and radius

Circle, centre and radius

Identifying the centre and radii of a circle

The fixed point where the compass needle rests is the centre of the circle. Let it be O.

Choose any three points A, B and C on the circle. By the definition of a circle,

OA = OB = OC.

The distance from the centre to any point on the circle is the radius. The segment joining the centre to that point is also called a radius.

If we use r for the radius, then

OA = OB = OC = r.

📚 Remember: The centre is fixed, and every point on the circle is at the same distance from it.

Circle and circumference

The circle is the closed curve consisting of points at a fixed distance from its centre. The total length around this curve is its circumference.

The circle together with its interior forms a circular region, also called a disc. Segments and sectors are parts of this region.

⭐ Chord and diameter

Circle, chord and diameter

A diameter is a chord through the centre

Chord

Choose two distinct points on the circle, such as A and B. Join them with the straight line segment AB.

A line segment joining two points on a circle is called a chord.

Thus, AB is a chord.

Many chords of different lengths can be drawn in the same circle.

Diameter

A chord that passes through the centre has a special name: diameter.

Suppose chord XY passes through the centre O. Then XY is a diameter.

Here,

XO = OY = r.

Therefore,

XY = XO + OY = r + r = 2r.

Hence,

d = 2r,

where d is the diameter's length and r is the radius.

📚 Important facts about chords and diameters

• Joining any two distinct points on a circle gives a chord.
• A chord through the centre is a diameter.
• Every diameter is a chord.
• Not every chord is a diameter.
• The diameter is twice the radius.
• A diameter is the longest chord of a circle.

⭐ Arc

Arcs of a circle

Arc, minor arc and major arc

Mark two distinct points X and Y on a circle. Travelling along the curved boundary from X to Y, there are two possible paths.

Each curved path is an arc of the circle.

Minor arc

When the endpoints are not the ends of a diameter, the shorter of the two arcs is the minor arc.

For example, we may refer to the shorter arc as arc XY.

Major arc

The longer arc between the same endpoints is the major arc.

Three letters are often used to identify a major arc clearly. If P lies on the longer arc, we can name it arc XPY.

A simple way to recognise them: For the same endpoints, the shorter curved path is the minor arc and the longer one is the major arc. If both arcs are equal, they are semicircular arcs.

⭐ Semicircle

Semicircle

A diameter divides the circle into two equal semicircular arcs

Now suppose X and Y are the endpoints of a diameter. The diameter divides the circular region into two equal halves.

The two arcs from X to Y have equal lengths. Each is half the circumference of the circle.

Each such arc is a semicircle, or semicircular arc. The region bounded by the arc and the diameter is a semicircular region.

Conversely, if the two arcs between the same endpoints have equal lengths, those endpoints are the ends of a diameter.

📚 Two important parts of a circular region

A circular region can be divided in different ways. Two commonly used parts are the segment and the sector.

⭐ Segment

Segment of a circle

A segment is bounded by a chord and an arc

A chord divides the circular region into two parts.

A segment is the region bounded by a chord and a corresponding arc.

Suppose AB is a chord. The chord AB and either of its corresponding arcs enclose a segment.

Minor segment

For a chord that is not a diameter, the smaller region between the chord and its minor arc is the minor segment.

Major segment

The larger region bounded by the same chord and its major arc is the major segment.

If the chord is a diameter, the two segments are equal semicircular regions, so neither is called minor or major.

⭐ Sector

Sector of a circle

Sector, minor sector and major sector

Draw the radii OA and OB from the centre O to points A and B on the circle.

These two radii and an arc between their endpoints enclose part of the circular region.

A sector is the region bounded by two radii and an arc between them.

Minor sector

The smaller region bounded by OA, OB and the minor arc AB is the minor sector.

Major sector

The larger region bounded by the same radii and the major arc AB is the major sector.

If the two radii form a diameter, they divide the circular region into two equal semicircular sectors. Neither is minor or major.

⭐ Distinguishing a segment from a sector

The terms “minor” and “major” apply when the two parts are unequal.

The difference in one line

Segment boundary = Chord + Arc.

Sector boundary = Two radii + Arc.

These are memory aids for the boundaries, not formulas for area.

📚 Parts of a circle at a glance

✍ Simple worked examples

✏️ Example 1: What is the diameter?

A circle has radius 6.5 cm. Find its diameter.

We know,

d = 2r.

Therefore,

d = 2 × 6.5 = 13 cm.

Answer: 13 cm.

✏️ Example 2: Which is a segment and which is a sector?

If a region is bounded by chord AB and a corresponding arc AB, it is a segment.

If a region is bounded by radii OA, OB and an arc AB, it is a sector.

⭐ Check your understanding

🔸 Can a circle have more than one chord?
Yes. Joining any two distinct points on the circle produces a chord.

🔸 Is every chord a diameter?
No. Only a chord through the centre is a diameter.

🔸 Into how many equal parts does a diameter divide a circular region?
Two semicircular regions.

🔸 What bounds a segment?
One chord and one arc.

🔸 What bounds a sector?
Two radii and one arc.

⚠️ Common mistakes and how to avoid them

• Do not confuse a chord with an arc. A chord is straight; an arc is curved.
• Remember that a diameter is a special chord. It passes through the centre.
• Do not confuse segments and sectors. A segment has a chord as its straight boundary; a sector has two radii.
• Use three letters when naming a major arc if needed. The middle letter identifies a point on the intended arc.
• Do not label equal halves as minor and major. A diameter gives two equal semicircular arcs or regions.

🧠 Key points to remember

🔹 All points on a circle are equally distant from its centre.

🔹 This fixed distance is its radius.

🔹 A segment joining two points on the circle is a chord.

🔹 A chord through the centre is a diameter.

🔹 d = 2r.

🔹 A diameter is the longest chord of a circle.

🔹 For endpoints that are not the ends of a diameter, the shorter arc is the minor arc and the longer arc is the major arc.

🔹 The endpoints of a diameter divide the circle into two equal semicircular arcs.

🔹 A chord and an arc bound a segment.

🔹 Two radii and an arc bound a sector.

📚 Quick revision

Next: Circle theorems

Once these basic ideas are clear, learn why exactly one circle passes through three non-collinear points and study the important chord theorems.

Next topic: Circle Theorems.