How this instrument works
The Inscribed Angle Theorem states that a central figure (measured from the circle's own center) is always exactly double any inscribed one subtending the same arc (measured from a point on the circle's remaining edge). The relationship holds regardless of exactly where on the circle's edge that second vertex sits — every such figure resting on a given arc shares the same measure, and that shared measure is always half the fixed central value.
This is one of the oldest results in circle geometry, traditionally credited in part to Thales of Miletus, and it has a striking special case: when the arc in question is a full semicircle (the central figure is 180°), every one resting on it must be exactly 90° — an inscribed triangle with one side as the circle's own diameter is always a right triangle, a fact sometimes called Thales' Theorem in its own right.
The theorem is the foundation several other circle results build from: the fact that opposite corners of a cyclic quadrilateral (one inscribed in a circle) always sum to 180° follows directly from it, since each pair of opposite corners ends up subtending arcs that together make up the whole circle.
- Enter the inscribed angle into the Inscribed angle field.
- Read Central angle: the sheet doubles the inscribed angle directly.
- Try 90° to see the semicircle special case, where the central figure reaches a full 180°.
Worked example — an inscribed angle of 30°
An inscribed angle of 30°, resting on some arc of a circle, corresponds to a central angle of exactly 60° subtending that same arc — the doubling relationship holding regardless of exactly where on the circle's remaining edge the inscribed angle's vertex sits.
An inscribed angle of 75° corresponds to a central angle of 150°, and if the arc were a full semicircle (central angle 180°), any inscribed angle resting on it would be exactly 90° — Thales' Theorem, the special case where an inscribed triangle with a diameter as one side is always a right triangle.
Questions
What is the Inscribed Angle Theorem?
It states that a central angle is always exactly double any inscribed angle subtending the same arc, regardless of where on the circle's edge that inscribed angle's vertex sits — one of the foundational results in circle geometry.
What is Thales' Theorem, and how is it related?
It's the special case of the Inscribed Angle Theorem where the arc is a full semicircle: since the central angle is then 180°, any inscribed angle resting on it must be exactly 90°, meaning a triangle inscribed in a circle with one side as the diameter is always a right triangle.
Do all inscribed angles on the same arc share the same measure?
Yes — every inscribed angle resting on a given arc, no matter where its vertex sits on the circle's remaining edge, shares the exact same measure, since each one is independently half of that arc's single, fixed central angle.
How does this relate to a cyclic quadrilateral's opposite angles?
A cyclic quadrilateral's opposite angles always sum to exactly 180°, a direct consequence of the Inscribed Angle Theorem — each pair of opposite angles subtends arcs that together make up the circle's full 360°, so their corresponding inscribed angles (each half their own arc) must sum to half of that, 180°.
Can the inscribed angle be obtuse?
Yes — an inscribed angle can range anywhere from just above 0° to just below 180°, and the central angle it corresponds to simply doubles that value, up to just below 360°, the full circle.