Ninth Grade Mathematics: Radius, Diameter, Chord, Arc & Sector
The circle is the most symmetrical and harmonious figure in plane geometry, representing the set of all points that lie at an exact, invariant distance from a fixed central reference point. In this chapter, you will master the foundational anatomy of the circle, exploring radii, diameters, chords, arcs, sectors, and segments.
The Formal Definition of a Circle
A circle is formally defined as the collection of all points in a plane that are at a constant, fixed distance from a fixed point in that same plane.
Anatomy of a Circle:
Circumference
. - ~ ~ - .
. ' | ' .
/ | \
/ | r \
; r v ;
| <----------O----------> | Diameter = 2*r
; * Center ;
\ /
\ Chord /
. _ *-------* '
' - ~ ~ - .
The fixed central point is called the center of the circle (commonly denoted by O). The fixed distance from the center to any point on the boundary curve is called the radius (plural: radii, denoted by r). The complete perimeter or boundary path enclosing the circular region is called the circumference.
Diameters and Chords
A chord of a circle is any straight line segment whose two endpoints both lie on the circle's circumference.
Chord vs. Diameter:
- Chord AB: Any segment connecting two points on the circle.
- Diameter CD: A special chord that passes directly through the center O.
The diameter (denoted by d) is the longest chord that can be drawn in any circle. Its length is exactly twice the radius:
Diameter = 2 * Radius (d = 2*r)
Every diameter bisects the circle into two identical halves, each called a semicircle.
Arcs: Minor, Major, and Semicircular
A continuous piece or portion of a circle's circumference lying between any two points is called an arc.
Major Arc APB
. - ~ ~ - .
P . ' ' .
/ \
/ \
; * O ;
| |
\ /
\ A B /
. _ * - - - - * _ '
Minor Arc AB
Two points A and B on a circle partition the circumference into two arcs: The minor arc is the shorter path between A and B (less than a semicircle). The major arc is the longer path between A and B, typically named using three letters (such as Arc APB) to indicate the direction of travel around the circle. If the endpoints A and B form a diameter, both arcs are semicircles.
Sectors and Segments
The circular region inside a circle is partitioned into two distinct geometric structures depending on the boundary curves used:
CIRCULAR SECTOR CIRCULAR SEGMENT
. - ~ ~ - . . - ~ ~ - .
. ' Major ' . . ' Major ' .
/ Sector \ / Segment \
; O ; ; O ;
| / \ | | |
\ r / \ r / \ /
\ *-----* / \ A B /
. _ A B _ ' . _ *-----------* _ '
Minor Sector Minor Segment
(Bounded by 2 Radii and an Arc) (Bounded by a Chord and an Arc)
A sector is the region bounded by two radii and the arc between them (resembling a slice of pie). A segment is the region bounded by a straight chord and the arc subtended by that chord.
Chapter Exercises
Exercise 1: Define the terms chord, diameter, and secant of a circle, explaining the geometric relationship and differences among them.
Exercise 2: A circle has a radius of 14 cm. Calculate its diameter, its circumference, and the length of a semicircle arc (use 22/7 for pi).
Exercise 3: Explain the essential structural difference between a circular sector and a circular segment, identifying the exact boundaries that enclose each region.
Solutions and Explanations
Solution 1: A chord is a straight line segment whose endpoints both lie on the circle. A diameter is a specialized chord that passes directly through the center of the circle; it is the longest possible chord in the circle, with length equal to twice the radius. A secant is an extended straight line that intersects the circle at two distinct points, cutting across the circle and extending infinitely in both directions (a chord is the finite segment of a secant line that lies inside the circle).
Solution 2: The radius is r = 14 cm. The diameter is d = 2r = 2(14) = 28 cm. The circumference is C = 2 * pi * r = 2 * (22/7) * 14 = 2 * 22 * 2 = 88 cm. A semicircle arc represents exactly half of the complete circumference: Semicircle Arc = C / 2 = 88 / 2 = 44 cm.
Solution 3: The essential difference between a sector and a segment lies in their boundary lines. A sector is enclosed by two straight radii that originate from the center of the circle, joined by the arc connecting their outer endpoints (it resembles a slice of pizza). A segment, by contrast, does not connect to the center point; it is enclosed entirely by a straight chord connecting two boundary points and the circular arc lying between those same two points.