Geometry: Circles Definitions and Properties

Study Notes

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.

Description

Explore the fundamental definitions and properties of circles in geometry. This quiz covers important concepts such as radius, diameter, chords, and secants, as well as calculations for circumference and area. Perfect for students looking to solidify their understanding of circular geometry.