Geometry: Circles Definitions and Properties

Definitions and Properties

• A circle is a two-dimensional geometric shape consisting of all points in a plane equidistant from a central point.
• The central point is called the center of the circle.
• The distance from the center to any point on the circle is called the radius.
• A line segment that passes through the center and has endpoints on the circle is called a diameter. The diameter is twice the radius.
• A line segment joining two points on the circle is called a chord.
• A chord passing through the center of the circle is a diameter.
• A secant is a line that intersects a circle at two points.

Calculating Circumference

• The circumference of a circle is the distance around the circle.
• The formula for calculating the circumference (C) of a circle is: C = 2πr, where 'r' is the radius.
• Alternatively, the circumference can be calculated using the diameter (d) as C = πd.
• The value of π (pi) is approximately 3.14159.

Calculating Area

• The area of a circle is the space enclosed by the circle.
• The formula for calculating the area (A) of a circle is: A = πr².
• 'r' represents the radius of the circle.

Arc Length

• An arc is a portion of a circle.
• Arc length is the distance along a section of the circumference.
• Arc length depends on the angle (in radians) subtended by the arc at the center of the circle.
• The formula for calculating arc length (s) is: s = rθ, where 'r' is the radius, and 'θ' is the angle in radians.

Sector Area

• A sector is a region bounded by two radii and an arc of a circle.
• The area of a sector (A) can be determined using the formula: A = (θ/2π) × πr², where 'θ' is the angle in radians and 'r' is the radius.
• Alternatively, for an angle measured in degrees, the area formula is: A = (θ/360) × πr².

Tangents

• A tangent to a circle is a line that intersects the circle at exactly one point.
• The tangent is perpendicular to the radius of the circle at the point of tangency.
• Two tangents drawn from an external point to a circle are always equal in length.

Sectors and Segments

• A sector is a region of a circle enclosed by two radii and an arc.
• A segment is a region of a circle bounded by a chord and an arc.
• Segment area can be calculated subtracting the area of a triangle from the sector area.

Concentric Circles

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

Circles in Coordinate Geometry

• In a coordinate plane, a circle can be described by an equation.
• The general equation for a circle with center (h, k) and radius 'r' is: (x - h)² + (y - k)² = r².

Circle Theorems

• Various theorems exist that define relationships between different elements of circles, such as chords, tangents, and arcs. Understanding these theorems is crucial for solving many geometry problems related to circles.
• The theorems describe relationships involving angles, segments, tangents, and other elements within/on circles.