Important Formulas: Circles | Mathematics (Maths) Class 9 PDF Download

• Area of a Circle Formula: The area occupied by a circle is determined by its radius → Area = πr², where 'r' is the radius and π ≈ 3.14.

• Circumference of a Circle Formula: The aggregate length of a circle's boundary is given by → Circumference = 2πr, where 'r' is the radius and π ≈ 3.14.

Important Theorems and Properties

Chords and Angles:

• Equal chords of a circle (or congruent circles) subtend equal angles at the center.
• If angles subtended by two chords at the center are equal, the chords are equal.
• The perpendicular drawn from the center to a chord bisects the chord.
• A line drawn from the center to bisect a chord is perpendicular to the chord.

Chords and Distance:

• Equal chords are equidistant from the center.
• Chords equidistant from the center are equal.

Arcs and Angles:

• Congruent arcs have equal chords, and vice versa.
• Congruent arcs subtend equal angles at the center.
• The angle subtended by an arc at the center is double the angle it subtends at any point on the remaining part of the circle.
• Angles in the same segment of a circle are equal.
• Angle in a semicircle is always a right angle (90°).

Cyclic Quadrilateral:

• The sum of opposite angles in a cyclic quadrilateral is 180°.
• If the sum of opposite angles in a quadrilateral is 180°, it is cyclic.

Converse & Construction:

• If a line segment joining two points subtends equal angles at two points on the same side, then all four points lie on a circle.
• One and only one circle passes through three non-collinear points.