Circles – Part 1
G.C.2 – Identify and describe relationships among
central, inscribed, and circumscribed angles.
Circles
A circle is a shape that is made up of all points
that are equidistant from a single point.
Circles have no corners and no sides.
Circles measure 360 degrees.
Circles
Radius – The length of the segment that connects
the center of a circle to any point on the circle.
Diameter – segment that passes through the
center and whose endpoints are both on the
circle.
Circumference – distance around a circle.
Formula is c = πd or c = 2πr
Area – the space inside a circle. Formula is A = πr2
Central Angles
Central Angles – Angle whose vertex is the
center of the circle. The measure of the
central angle is equal to the measure of the
arc it intercepts on the circle.
X = arc AB
X = 80
Inscribed Angles
Inscribed angle – angle in a circle whose vertex is on
the circle. The measure of the inscribed angle is
exactly one half the measure of the intercepted arc.
Likewise, an intercepted arc is exactly two times the
measure of an inscribed angle.
X = ½(arc AC)
X = ½(100)
X = 50
Arcs
Arc – a segment of the curve on a circle.
Major arc – arc that measures more than 180
degrees.
Minor arc – arc that measures less than 180
degrees.
Intercepted arc – arc whose endpoints are
also endpoints of segments that form an
angle inside, outside, or on the circle.
Tangent Angles
Tangent Angle – Angle whose vertex is on the circle
and is formed by a tangent and a chord. The measure
of a tangent angle is equal to one half the measure
of its intercepted arc.
X = ½(min.arc AB)
X = ½(120)
X = 60
Other Angles
Angles created by two secants or tangents with the
vertex outside the circle – In this case, there will be two
intercepted arcs created. The measure of the angle is
equal to one half the DIFFERENCE between the
measures of the two intercepted arcs.
X = ½(arc AE – arc BD)
X = ½(80 – 20)
X = ½(60)
X = 30
Other Angles
This is an example of an angle created by two tangents.
The same formula applies here, as well.
X = ½(maj.arc AC – min.arc AC)
X = ½(260– 100)
X = ½(160)
X = 80
Other Angles
Angle formed inside the circle by two chords – In this
case, the measure of an angle (and its vertical angle) is
equal to one half the SUM of the two intercepted arcs.
X = ½(min.arc AC + min.arc BD)
X = ½(170+ 70)
X = ½(240)
X = 120
Segments
Tangent – A line that intersects a circle in only
one point. The tangent is always
perpendicular to the radius.
Secant – A line or line segment that intersects
a circle in two points.
Chord – A line segment whose endpoints are
on a circle.
Diameter – a type of chord that passes
through the center of the circle.
Segments