Chapter 11: Circles
Section 11.1: Parts of a Circle
Circle – the set of all points on a plane equidistant from a given point called the center
(1) radius – a segment whose endpoints are the center of the circle and a point on the circle
(2) chord – a segment whose endpoints are both on the circle
(3) diameter – a chord that contains the center of the circle
(4) secant – a line that intersects the circle at two points
(5) tangent - a line in the plane of the circle that intersects the circle at only one point
secant
chord
diameter
radius
tangent
Theorem 11.1 All radii of a circle are congruent.
Theorem 11.2 The measure of the diameter of a circle equals twice the length of the radius. d  2r or
1
r d
2
Section 11.2: Arcs and Central Angles
Central angle – an angle whose vertex is the center of the circle and whose sides are contained by two
radii APB
Arcs – the set of points on a circle whose endpoints are two endpoints on the circle
(1) minor arc – the arc in the interior of a given central angle AB
(2) major arc – the arc in the exterior of a given central angle AOB
(3) semicircle – the arc whose endpoints are the endpoints of a diameter SOT or SAT  SBT
S
T
Arc Measure
(1) minor arc – equals the measure of its central angle mAB  mAPB
(2) major arc – equals 360 minus the measure of its central angle mAOB  360  mAPB
(3) semicircle = equals 180 mSOT  mSAT  mSBT  180
Adjacent arcs – two nonoverlapping arcs on a circle that share a common endpoint AB and BT
Postulate 11.1 (Arc Addition Postulate) The sum of the measures of two adjacent arcs is the measure
of the arc formed by the adjacent arcs. mAB  m BT  mAT
Theorem 11.3 In a circle or in congruent circles, two minor arcs are congruent if and only if their
corresponding central angles are congruent.
Section 11.3: Arcs and Chords
Theorem 11.4 In a circle or congruent circles, two minor arcs are congruent if and only if their
corresponding chords are congruent.
If AB  CD, then AB  CD.
Or
If AB  CD, then AB  CD.
Theorem 11.5 In a circle, a diameter bisect a chord and its arc if and only if it is perpendicular to the
chord.
If diameter DC bisects AB and AB, then DC  AB.
If diameter DC  AB, then DC bisects AB and AB.
Arc Measure
Arc Length
major arc: 360 degrees - the length of the circumference
the degree measure of
proportional to the measure of the central
its central angle
angle when compared to the entire circle.
minor arc: the degree
measure of the central
angle
Theorem - In the same or in congruent circles, two arcs are congruent if and only if their central angles
are congruent.
Arc of a chord - when a minor arc and a chord have the same endpoints.
Theorem - In a circle or congruent circles, two minor arcs are congruent if and only if their
corresponding chords are congruent.
Chord AD is congruent to chord BC, therefore Arc AD is congruent to Arc BC
Example 1 Radius Perpendicular to a Chord
Circle N has a radius of 36.5 cm. Radius NH is perpendicular to chord FG , which is 53 cm long.
a. If m FG = 85, find m HG .
1
NH bisects FG , so m HG = 2m FG .
1
m HG = 2m FG
1
m HG = 2(85) or 42.5
b. Find NZ.
Definition of arc bisector
m FG = 85
Draw radius NG . NZG is a right triangle.
NG = 36.5
r = 36.5
NH bisects FG .
A radius perpendicular to a chord bisects it.
1
ZG = 2(FG)
1
= 2 (53) or 26.5
Definition of segment bisector
FG = 53
Use the Pythagorean Theorem to find NZ.
(NZ)2 + (ZG)2 = (NG)2
Pythagorean Theorem
2
2
2
(NZ) + (26.5) = (36.5)
ZG = 26.5, NG = 36.5
(NZ)2 + 702.25 = 1332.25
Simplify.
(NZ)2 = 630
Subtract 702.25 from each side.
NZ ≈ 25.1
Take the square root of each side.
Example 2 Chords Equidistant from Center
Chords FG and LY are equidistant from the center. If the radius of M is 32,
find FG and BY.
FG and LY are equidistant from M, so FG  LY .
Draw FM and LM to form two right triangles.
Use the Pythagorean Theorem.
(FN)2 + (NM)2 = (FM)2
Pythagorean Theorem
(FN)2 + (22)2 = (32)2
NM = 22, FM = 32
2
(FN) + 484 = 1024
Simplify.
(FN)2 = 540
Subtract 484 from each side.
FN ≈ 23.2
Take the square root of each side.
1
FN = 2(FG), so FG ≈ 2(23.2) or 46.4.
1
1
FG  LY , so LY also equals 46.4. BY = 2LY, so BY ≈ 2(42.4) or 23.2.
Section 11.4: Inscribed Polygons
Inscribed polygon = Circumscribed circle - a polygon is inscribed in a circle or a circle is
circumscribed about a polygon if and only if every vertex of the polygon lies on the circle
Pentagon ABCDE is inscribed the circle
Circumscribed polygon = Inscribed circle – a polygon is circumscribed about a circle or a circle is
inscribed in a circle if and only if each side is tangent to the circle
Pentagon PQRST is circumscribed about the circle
P
A
T E
B
Q
D
S
C
R
Theorem 11.6 In a circle or in congruent circles, two chords are congruent if and only if they are
equidistant from the center.
If WX  YZ , then CL  CM .
or
If CL  CM , then WX  YZ .
Section 11.5: Circumference of a Circle
Circumference – the distance around a circle
Theorem 11.7 (Circumference of a Circle) The circumference of a circle equals twice the product of 
and the radius or it equals the product of  and the diameter. C  2 r   d

C C

2r d
Arc length – the linear length of an arc; a fraction of the circumference of a circle
s
m
2 r , m  measure of the arc
360
A circle is the collection of points equidistant from a given point, called the center. A circle is named after
its center point. The distance from the center to any point on the circle is called the radius, (r), the most
important measurement in a circle. If you know a circle’s radius, you can figure out all its other
characteristics. The diameter (d) of a circle is twice as long as the radius (d = 2r) and stretches between
endpoints on the circle, passing through the center. A chord also extends from endpoint to endpoint on the
circle, but it does not necessarily pass through the center. In the figure below, point C is the center of the
circle, r is the radius, and AB is a chord.
Central Angles
An angle whose vertex is the center of the circle is called a central angle.
The degree of the circle (the slice of pie) cut by a central angle is equal to the measure of the angle. If a
central angle is 25º, then it cuts a 25º arc in the circle.
Circumference of a Circle
The circumference is the perimeter of the circle. The formula for circumference of a circle is
where r is the radius. The formula can also be written C = πd, where d is the diameter. Try to find the
circumference of the circle below:
Plugging the radius into the formula, C = 2πr = 2π (3) = 6π.
Arc Length
An arc is a part of a circle’s circumference. An arc contains two endpoints and all the points on the circle
between the endpoints. By picking any two points on a circle, two arcs are created: a major arc, which is by
definition the longer arc, and a minor arc, the shorter one.
Since the degree of an arc is defined by the central or inscribed angle that intercepts the arc’s endpoints,
you can calculate the arc length as long as you know the circle’s radius and the measure of either the
central or inscribed angle.
The arc length formula is
where n is the measure of the degree of the arc, and r is the radius.
Circle D has radius
9. What is the length
of arc AB?
In order to figure out the length of arc AB, you need to know the radius of the circle and the measure of
, the inscribed angle that intercepts the endpoints of AB. The question tells you the radius of the circle, but
it throws you a little curveball by not providing you with the measure of
. Instead, the question puts
in a triangle and tells you the measures of the other two angles in the triangle. Like we said, only a little
curveball: You can easily figure out the measure of
because, as you (better) know, the three angles of a
triangle add up to 180º.
Since angle c is an inscribed angle, arc AB must be 120º. Now you can plug these values into the formula
for arc length:
Section 11.6: Area of a Circle
Theorem 11.8 (Area of a Circle) The area of a circle equals the product between  and the square of
the radius.
A   r2
Sector of a circle – the region of a circle bounded by a central angle and its corresponding arc
Sector
Theorem 11.9 (Area of a Sector of a Circle) The area of a sector of a circle is fraction of the area of
the circle.
A
m
 r 2 where m  measure of the arc or central angle
360