An inscribed angle equals half its intercepted arc — one of the most useful facts in circle geometry.
The Mnemonic
Half the Arc, Every Time
An inscribed angle is an angle formed by two chords that share an endpoint on the circle. The Inscribed Angle Theorem states: an inscribed angle is always exactly HALF the measure of its intercepted arc (the arc "cut off" between the angle's two other endpoints).
Since a central angle (an angle with its vertex at the circle's center) is always EQUAL to its intercepted arc's measure, this also means: an inscribed angle is always half of the central angle that intercepts the same arc.
Both angles intercept the same arc — the inscribed angle is always exactly half the central angle
💡 Memory Trick
"An inscribed angle equals half its intercepted arc." Vertex ON the circle = inscribed angle = half the arc. Vertex AT the center = central angle = equal to the arc. The inscribed angle is always the "discount" version — half of what the central angle gives you for the exact same arc.
Why It Works
A Proof Using Isosceles Triangles
One clean way to see this: draw a radius from the circle's center to the inscribed angle's vertex. This creates two isosceles triangles (since two sides of each are radii, and all radii are equal) sharing that center point. By the Isosceles Triangle Theorem, each triangle's base angles are equal.
Working through the angle relationships (using the Exterior Angle Theorem on each small triangle) shows that the inscribed angle always comes out to exactly half of the central angle intercepting the same arc — this is provable rigorously, not just an observed pattern, and it holds true no matter where on the circle the inscribed angle's vertex sits (as long as it intercepts the same arc).
Using It In A Proof
The Semicircle Special Case
One especially useful consequence of this theorem deserves its own name: any inscribed angle that intercepts a semicircle (an arc of exactly 180°, meaning the two other endpoints form a diameter) is ALWAYS a right angle.
1
Identify the intercepted arc
Determine which arc the inscribed angle "cuts off" between its two non-vertex points.
If the two non-vertex points are the endpoints of a diameter, the arc is 180°, making the inscribed angle automatically 90° — a right angle, every single time.
Full Worked Example
Using the Semicircle Case to Find a Missing Angle
Given: Triangle ABC is inscribed in a circle, where AC is a diameter. ∠A = 35°. Find: ∠B and ∠C.
1
Identify the semicircle case
Since AC is a diameter, ∠B (the inscribed angle at vertex B, intercepting the semicircle arc AC) must be 90°.
2
Use the Triangle Angle Sum Theorem
∠A + ∠B + ∠C = 180°, so 35° + 90° + ∠C = 180°.
3
Solve for ∠C
∠C = 180° − 125° = 55°.
4
Confirm using the Inscribed Angle Theorem directly
∠A intercepts arc BC, so arc BC = 2×35° = 70°. ∠C intercepts arc AB, so arc AB = 2×55° = 110°. Arc BC + Arc AB should equal the semicircle: 70° + 110° = 180°. ✓
This cross-check using arcs directly confirms the triangle-angle-sum approach gave the correct answer.
🎯 Quick Worked Example
An inscribed angle intercepts an arc of 86°. Find the inscribed angle's measure.
1
Apply the theorem. Inscribed angle = arc / 2.
2
Calculate. 86° / 2 = 43°.
3
Conclude. The inscribed angle measures 43°.
📌 Exam Application
The semicircle special case (any triangle inscribed with one side as a diameter has a 90° angle opposite that diameter) is one of the most frequently tested facts in circle geometry — it appears constantly disguised inside larger problems, letting you instantly identify a right angle without any calculation.
⚠️ Most Common Inscribed Angle Theorem Mistakes
Trap 1 — Doubling instead of halving (or vice versa): The inscribed angle is HALF the arc; the arc is DOUBLE the inscribed angle — mixing up which direction to multiply or divide is a very common error.
Trap 2 — Confusing an inscribed angle with a central angle: A central angle's vertex is AT the circle's center and equals its arc directly; an inscribed angle's vertex is ON the circle and equals HALF its arc — using the wrong relationship for the wrong vertex location gives an answer twice (or half) the correct one.
✓ Quick Self-Test
1) State the Inscribed Angle Theorem. 2) What is special about an inscribed angle that intercepts a semicircle? 3) An inscribed angle intercepts a 120° arc — find the angle. 4) Triangle XYZ is inscribed in a circle with XZ as a diameter, and ∠X = 50° — find ∠Y and ∠Z. 5) How does a central angle's relationship to its arc differ from an inscribed angle's relationship to the same arc?