Formulas for Angles in
Circles
Formed by Radii, Chords,
Tangents, Secants
There are basically five circle
formulas that
you need to remember…..
1. Central Angles
A central angle is an angle formed by two intersecting radii
such that its vertex is at the center of the circle.
FORMULA
Outside arc = Inside angle
m AOB  mAOB
m AOB  80
2. Inscribed Angle:
• An inscribed angle is an angle with its vertex "on" the
circle, formed by two intersecting chords
FORMULA
1
• Outside arc =
Inside Angle
2
1
m ABC  AC
2
1
m ABC  (100)
2
m x  50
Special situation
• An angle inscribed in a semi-circle is a
right angle.

  
1
1
m ABC  m AC  180  90
2
2
3. Tangent Chord Angle
• An angle formed by an intersecting tangent and chord
has its vertex "on" the circle.
1
Tangent Chord Angle = Intercepted Arc
2
1
m ABC  m AB
2
1
m ABC  (120)
2
m<ABC = 60º
4. Angle Formed Inside of a Circle by
Two Intersecting Chords
• When two chords intersect "inside" a circle, two pairs of
vertical angles are formed. Remember: vertical angles
are equal.
FORMULA
Angle Inside = ½ the SUM of Intercepted arcs
1
m BED  (mAB  mBD)
2
1
m x  (70  170)
2
m x  120
5. Angle Formed Outside of a
Circle by the Intersection of …..
a.) Two Tangents
b.) Two Secants
c.) Tangent and a Secant
FORMULA FOR ALL THREE OPTIONS…
1
Angle Formed Outside =
Difference of Intercepted Arcs
2
a.) Two Tangents
• <ABC is formed by two tangents intersecting outside of
circle O.
• X = half the difference in the two measurements
major
minor


1
m ABC   m AC  m AC 

2 

1
m ABC   260  100 
2
m ABC  80
b.) Two Secants
• <ACE is formed by two secants intersecting outside of
circle O.
• X = Half the difference in the two measurements
l arg er
smaller


1
m ABE   m AE  mBD 

2 

1
m ABC   80  20 
2
m ABC  30
c.) Tangent and Secant
• <ABD is formed by a tangent and a secant
intersecting outside of circle O.
• X = Half the difference in the two measurements
l arg er
smaller


1
m ABD   m AD  m AC 

2 

1
m ABD  100  30 
2
m ABD  35