The formula that unifies all circle angle problems — depending only on WHERE the vertex sits.
The Mnemonic
One Big Rule, Sorted by Vertex Location
Every angle formed by chords, secants, or tangents relative to a circle follows one of two patterns, determined entirely by WHERE the angle's vertex sits: Vertex ON the circle (an inscribed angle): equals HALF the intercepted arc — this is just the Inscribed Angle Theorem, already covered.
Vertex INSIDE the circle (two chords crossing): equals HALF the SUM of the two intercepted arcs (the arc "in front of" the angle, plus the arc directly across from it). Vertex OUTSIDE the circle (two secants, a secant and tangent, or two tangents): equals HALF the DIFFERENCE of the two intercepted arcs (the far arc minus the near arc).
Angle at external point P = half the DIFFERENCE of the two intercepted arcs (far arc minus near arc)
💡 Memory Trick
"The formula that unifies all circle angle problems." ON the circle: half the SINGLE arc. INSIDE: half the SUM of two arcs. OUTSIDE: half the DIFFERENCE of two arcs. Notice the pattern: as the vertex moves from on, to inside, to outside the circle, the formula shifts from one arc, to adding two arcs, to subtracting two arcs.
Why It Works
Both New Cases Come From the Inscribed Angle Theorem
The INSIDE case (two chords crossing) can be shown by drawing one extra chord connecting two of the four endpoints, creating a triangle where the crossing angle is an EXTERIOR angle. By the Exterior Angle Theorem, that exterior angle equals the sum of the two remote interior angles — and each of those interior angles is itself an inscribed angle (half its own arc). Adding half of one arc plus half of the other arc gives exactly half the SUM of both arcs.
The OUTSIDE case works similarly but uses subtraction instead of addition: connecting the near intersection points creates a triangle where the outside angle is one of the triangle's OWN interior angles (not exterior), related to the two inscribed angles through subtraction rather than direct addition — the algebra works out to exactly half the DIFFERENCE of the two arcs. Both derivations lean on the same core tool (Inscribed Angle Theorem) applied inside a cleverly constructed triangle.
Using It In A Proof
Identifying the Correct Case
The entire skill in these problems is correctly locating the vertex and identifying BOTH intercepted arcs before choosing sum or difference.
1
Locate the vertex
On the circle → inscribed angle (single arc). Inside the circle → sum case. Outside the circle → difference case.
2
Identify both intercepted arcs
For the inside/outside cases, find the two DIFFERENT arcs the angle (and its vertical angle, if inside) actually cuts off — these are usually the "far" arc and "near" arc relative to the vertex.
3
Apply the matching formula
Half the sum (inside) or half the difference, far arc minus near arc (outside).
Full Worked Example
Solving an Outside-Vertex Problem
Given: Two secants from an external point intercept arcs of 100° (far) and 30° (near). Find: the angle at the external point.
1
Confirm the vertex location
The vertex is outside the circle (where the two secants meet) — this is the difference case.
2
Set up the formula
Angle = ½(far arc − near arc) = ½(100° − 30°).
3
Calculate
Angle = ½(70°) = 35°.
4
Compare with the inside case for contrast
If this SAME pair of arcs (100° and 30°) had instead been formed by chords crossing INSIDE the circle, the angle would be ½(100°+30°) = 65° instead — a genuinely different answer, showing why vertex location matters so much.
This comparison highlights exactly why identifying vertex location correctly is the single most important step in these problems.
🎯 Quick Worked Example
Two chords cross inside a circle, intercepting arcs of 80° and 40°. Find the angle at the intersection point.
1
Confirm the vertex location. Inside the circle (chords crossing) — this is the sum case.
2
Apply the formula. Angle = ½(80° + 40°).
3
Calculate. Angle = ½(120°) = 60°.
📌 Exam Application
The single fastest way to avoid errors on these problems is to explicitly state out loud (or in your notes) whether the vertex is on, inside, or outside the circle BEFORE writing any formula — jumping straight to a half-sum-or-difference calculation without this check is the most common source of using the wrong operation.
⚠️ Most Common Angles Formed by Secants and Tangents Mistakes
Trap 1 — Using sum instead of difference (or vice versa): Inside the circle uses SUM; outside uses DIFFERENCE — since both formulas involve the same 'half of two arcs' structure, mixing up which operation applies to which vertex location is extremely common.
Trap 2 — Misidentifying which arc is 'far' and which is 'near' in the outside case: The difference must always be FAR arc minus NEAR arc (giving a positive result) — subtracting in the wrong order gives a negative angle, which is a clear sign the arcs were mislabeled.
✓ Quick Self-Test
1) What formula applies when the vertex is ON the circle? 2) What formula applies when the vertex is INSIDE the circle? 3) What formula applies when the vertex is OUTSIDE the circle? 4) Two secants from an external point intercept arcs of 150° and 50° — find the angle at that point. 5) Why does the 'inside' case use addition while the 'outside' case uses subtraction?