Central and Inscribed Angles
An inscribed angle is half the central angle on the same arc.
Key idea
An inscribed angle is half the central angle on the same arc.
In depth
An inscribed angle is half the central angle on the same arc. So all inscribed angles on the same arc are equal, and an angle in a semicircle is 90°. This follows from isosceles triangles formed by radii.
Worked example
A central angle of 100° means the inscribed angle on the same arc is 50°.
Real-world uses
- Designing arches and circular stadium layouts
- Geometry proofs in exams
Common mistake
Comparing angles that stand on different arcs.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Central and Inscribed Angles?
An inscribed angle is half the central angle on the same arc. It is a topic at the advanced level in Geometry.
Where is Central and Inscribed Angles used in real life?
Real-world examples: Designing arches and circular stadium layouts; Geometry proofs in exams.
What should I learn before Central and Inscribed Angles?
Make sure you are comfortable with Sector Area, the previous topic in Geometry.
What is a common mistake with Central and Inscribed Angles?
Comparing angles that stand on different arcs.
How can I practise Central and Inscribed Angles?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Central and Inscribed Angles?
Move on to Chord and Tangent Properties next.
Related advanced topics in Geometry
- Heron’s Formula – Triangle area from three sides.
- Regular Polygon Area – Use apothem and perimeter.
- Arc Length of a Circle – Part of the circumference.
- Sector Area – Part of the circle’s area.
- Chord and Tangent Properties – A tangent is perpendicular to the radius at the point of contact.
- Cones – Volume and surface area of a cone.