Circle Chapter Notes | Mathematics Class 6 ICSE PDF Download

Introduction

Imagine drawing a perfect round shape, like the sun or a coin, where every point on the edge is equally far from one special point in the middle. That’s a circle! Circles are fascinating because they are simple yet appear everywhere—in wheels, clocks, and even pizza! This chapter will take you on a journey to understand the different parts of a circle, how they work together, and how to draw them using a compass and ruler. Get ready to explore this beautiful shape step by step!

Circle

• A circle is a round shape where all points on the edge are at the same distance from a fixed point called the center.
• The area inside the circle is called the circular region.
• It is a closed curve formed in a plane.
• Example: In a figure, point O is the center, and the shaded area inside the circle is the circular region.

Parts of a Circle

Centre

• The center is the fixed point inside the circle from which all points on the edge are equally distant.
• It is the heart of the circle, defining its position.
• Example: In a figure, point O is the center of the circle.

Radius

• The radius is the distance from the center to any point on the circle’s edge.
• It is the fixed length that defines the size of the circle.
• Example: In a figure, line segment OA is a radius of the circle.
• Solved Example: If the radius of a circle is 4 cm, the distance from the center O to point P on the circle is 4 cm.

Chord

• A chord is a line segment that connects any two points on the circle’s edge.
• It lies completely inside the circle.
• Example: In a figure, line segment PQ is a chord of the circle.
• Solved Example: If points A and B lie on a circle and are joined by a line segment AB, then AB is a chord.

Segment

• A chord divides the circle into two parts, each called a segment.
• The larger part (more than half the circle) is the major segment.
• The smaller part (less than half the circle) is the minor segment.
• The major segment includes the center of the circle.
• Points to Remember:
  • The major segment contains the center.
  • A diameter splits the circle into two equal segments.
  • A quarter of a circle, bounded by two perpendicular radii, is called a quadrant.

Diameter

• The diameter is a special chord that passes through the center of the circle.
• It is the longest chord in a circle.
• Formula: Diameter = 2 × Radius
• It divides the circle into two equal parts, each called a semicircle.
• Example: In a figure, line segment AB is a diameter passing through the center O.
• Solved Example: If the radius of a circle is 6 cm, the diameter is 2 × 6 = 12 cm.

Circumference

• The circumference is the total length of the circle’s boundary.
• It represents the distance around the circle.
• Example: In a figure, PQR is the circumference of the circle.

Arc

• An arc is a portion of the circumference between two points on the circle.
• If the arc is longer than half the circumference, it is a major arc.
• If the arc is shorter than half the circumference, it is a minor arc.
• Example: In a figure, arc PAQ is a minor arc, and arc PBQ is a major arc.

Sector

• A sector is the region enclosed by two radii and an arc of the circle.
• A minor sector is formed by a minor arc and two radii.
• A major sector is formed by a major arc and two radii.
• Example: In a figure, OAPB is a minor sector formed by minor arc APB and radii OA and OB. OAQB is a major sector formed by major arc AQB and radii OA and OB.

Interior and Exterior

• The interior of a circle is the set of all points inside the circle.
• The exterior of a circle is the set of all points outside the circle.
• Example: In a figure, points O, A, B, and P are inside the circle (interior), while points C, Q, and D are outside (exterior).
• Solved Example: If the radius of a circle is 5 cm and a point P is 4 cm from the center, P lies in the interior. If a point Q is 6 cm from the center, Q lies in the exterior.

Secant

• A secant is a straight line that intersects the circle at two distinct points.
• It crosses through the circle, touching it at two places.
• Example: In a figure, line AB is a secant intersecting the circle at points C and D.

Tangent

• A tangent is a straight line that touches the circle at exactly one point.
• The point where the tangent touches the circle is called the point of contact.

Example: In a figure, line PQ is a tangent touching the circle at point R, the point of contact.

Points to Remember:

• A circle has only one center but can have infinite radii, chords, and diameters.

Construction of Circles

To Construct a Circle of Given Radius

• Measure the desired radius using a compass and ruler.
• Mark the center point on the paper.
• Place the compass’s pointed end at the center.
• Rotate the compass to draw the circle.

Example:To draw a circle with center O and radius 5 cm:

• Step 1: Use a ruler to set the compass to 5 cm.
• Step 2: Mark point O as the center.
• Step 3: Place the compass point at O.
• Step 4: Rotate the compass to draw the circle.

To Construct a Circle with Given Line Segment as a Diameter

• Find the midpoint of the line segment by drawing its perpendicular bisector.
• Use the midpoint as the center and half the line segment’s length as the radius.
• Draw the circle using a compass.

Example: To draw a circle with line segment AB of length 8 cm as the diameter:

• Step 1: Draw the perpendicular bisector of AB, intersecting at point O.
• Step 2: Use point O as the center and draw a circle with radius AO or OB (4 cm).

To Construct a Circle Passing Through Three Non-Collinear Points

• Join two pairs of the three points to form two chords.
• Draw perpendicular bisectors of these chords; their intersection is the center.
• Use the distance from the center to any of the three points as the radius.
• Draw the circle using a compass.

Example: To draw a circle passing through points A, B, and C:

• Step 1: Join AB and AC, then draw their perpendicular bisectors, intersecting at point O.
• Step 2: Draw a circle with center O and radius OA, OB, or OC.