Circles: Radius, Diameter, and Chords - Sixth Grade Mathematics
A circle is a perfectly symmetrical, closed two-dimensional figure consisting of all points in a plane that are equidistant from a fixed central point.
Anatomy of a Circle
Unlike polygons, circles have no straight sides or vertices. Every circle is defined completely by its center and its distance from that center.
Anatomy of a Circle:
Point on Circle
*
.-'|'-.
.-' | '-.
.' | '.
/ Radius \
; | ;
| * |
-----|-------- Center -----|----- <- Secant Line
| * |
; / \ ;
\ / \ /
'. / \ .'
.-*-------*-' <- Chord
'-...-'
Tangent Line (touches at 1 point)
Core Circle Terminology:
1. Center: The central point equidistant from all points along the curved perimeter.
Circles are named by their center: Circle O.
2. Radius (r): Any line segment connecting the center to any point on the circle.
Plural is radii. All radii of a given circle are congruent.
3. Diameter (d): A straight line segment passing directly through the center whose endpoints
both lie on the circle. It is the longest chord possible in a circle.
4. Chord: Any line segment connecting two points on the circle, whether it passes
through the center or not.
5. Tangent: A straight line that touches the circle at exactly one point.
6. Secant: A straight line that passes through the circle, cutting it at two points.
The Fundamental Radius-Diameter Relationship
Because a diameter passes through the center point, it is composed of two collinear radii joined end-to-end.
Mathematical Relationships:
Diameter = 2 * Radius (d = 2r)
Radius = Diameter / 2 (r = d / 2)
For example, if a circle has a radius of 7.5 cm, its diameter is 2 * 7.5 = 15 cm. Conversely, if a circle has a diameter of 22 inches, its radius is 22 / 2 = 11 inches.
Arcs and Central Angles
The curved boundary of a circle can be divided into portions called arcs.
Arcs and Central Angles:
B
.-*'-.
.-' | '-.
.' | '.
/ | \
; Central ;
| Angle |
*---------* O |
A Radius '. .'
'-...-'
Arc AB is the curved path along the circle connecting point A and point B.
Central Angle AOB is the angle whose vertex is at the center O.
Arc Classifications:
1. Minor Arc: An arc smaller than a semicircle (less than 180°). Named with 2 letters: Arc AB.
2. Semicircle: An arc that equals exactly half of a circle (180°). Its endpoints are the diameter.
3. Major Arc: An arc larger than a semicircle (greater than 180°). Named with 3 letters: Arc ACB.
The measure of an arc equals the measure of its central angle:
m(Arc AB) = m(Central Angle AOB).
Chords and Concentric Circles
A chord connects any two points on a circle. Every diameter is a chord, but not every chord is a diameter (only chords passing through the center are diameters).
Concentric circles are circles that share the exact same center point but have different radii, resembling the rings of a target or ripples in a pond.
Concentric Circles (Same Center O, Different Radii r1 and r2):
.-'''''-.
.-' .-. '-.
/ / * \ \
; ( O ) ;
\ \ / /
'-. '-' .-'
'-.....-'
Exercises
Exercise 1. Complete each statement: (a) If the radius of a circle is 9.4 cm, its diameter is _; (b) If the diameter of a bicycle wheel is 26 inches, its radius is ; (c) The longest possible chord in any circle is called the _; (d) All radii of the same circle are .
Exercise 2. A circle has points A, B, and C on its perimeter, with center point O: (a) Segment OA is a _; (b) Segment AB passing through center O is both a and a _; (c) Segment BC not passing through center O is a .
Exercise 3. In Circle P, a central angle Angle QPR measures 75 degrees. (a) What is the degree measure of minor arc QR? (b) What is the degree measure of major arc QSR?
Exercise 4. A circular dartboard has a bullseye at the center. The inner red circle has a radius of 2 cm, while the outer scoring ring has a diameter of 30 cm. What is the radius of the outer ring, and how much longer is its radius than the inner circle's radius?
Solutions
Solution to Exercise 1. For (a), diameter = 2 * radius = 2 * 9.4 = 18.8 cm. For (b), radius = diameter / 2 = 26 / 2 = 13 inches. For (c), the longest possible chord in any circle is called the diameter. For (d), all radii of the same circle are congruent (equal in length).
Solution to Exercise 2. For (a), Segment OA connects the center to a point on the circle, so it is a radius. For (b), Segment AB connects two points on the circle and passes through the center, so it is both a chord and a diameter. For (c), Segment BC connects two points on the circle without passing through the center, so it is a chord.
Solution to Exercise 3. For (a), the degree measure of a minor arc equals the measure of its central angle, so m(minor arc QR) = 75 degrees. For (b), a full circle contains 360 degrees. The major arc QSR covers the remainder of the circle: 360 - 75 = 285 degrees.
Solution to Exercise 4. The outer scoring ring has a diameter of 30 cm, so its radius is 30 / 2 = 15 cm. The difference between the radius of the outer ring (15 cm) and the radius of the inner red circle (2 cm) is 15 - 2 = 13 cm.