Central and Inscribed Angles
An inscribed angle is half the central angle on the same arc.
About this topic
A central angle has its vertex at the circle's centre, and its measure equals the arc it cuts off. An inscribed angle has its vertex on the circle and measures half of its intercepted arc. So an angle inscribed in a semicircle is always 90°.
Inscribed angles on the same arc are equal. In a cyclic quadrilateral, opposite angles add up to 180°. These theorems help find unknown angles in circle diagrams and prove other results.
Worked example: an arc measures 80°. A central angle that cuts it is 80°, while an inscribed angle on that arc is 40°. If a triangle is inscribed in a circle with one side as the diameter, the angle opposite is 90°. In a cyclic quadrilateral with one angle 75°, the opposite angle is 105°. If two inscribed angles stand on the same arc, both measure the same, perhaps 35°.
These theorems are used in designing arches, stadiums and optical devices, and they help find the centre of a circle by drawing two right angles on diameters. The main mistake is mixing up the rules: the central angle equals its arc, but the inscribed angle is half. Another is applying the 180° rule to quadrilaterals that do not have all four vertices on the circle. Always find which arc each angle intercepts, and mark equal arcs and angles on the diagram.
Key ideas
- Central angle equals its arc
- Inscribed angle = half the arc
- Angles in a semicircle are 90°
Quick check
Use cases: where Central and Inscribed Angles helps
- Homework: review the key idea behind Central and Inscribed Angles before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Central and Inscribed Angles without notes.
- Visual learning: play with the interactive lab to see how Central and Inscribed Angles changes when you adjust its inputs.
- Teaching a friend: explain Central and Inscribed Angles in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Central and Inscribed Angles?
An inscribed angle is half the central angle on the same arc.
What will I learn in Central and Inscribed Angles?
Central angle equals its arc. Inscribed angle = half the arc. Angles in a semicircle are 90°.
Is Central and Inscribed Angles part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Central and Inscribed Angles lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Central and Inscribed Angles?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
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