Circle Theorems

Circle Theorems Maths: A short guide to Circle Theorems Geometry

Circle theorems might sound like a complex topic, but when you boil them down, they’re simply the rules governing everyday round things we encounter – from the coins in our pockets, to the wheels on the morning bus, to the pizza we enjoy for dinner! 

Before we go through this fascinating part of Geometry, remember that at Cazoom Maths, we have lots of Circle Theorem worksheets for you. You can see some examples below and find all those Circle Theorem worksheets here. 

1. Understanding Circle Theorems

At their core, circle theorems provide insights about angles and lengths within circles. These insights have been refined and created over centuries, offering a mathematical perspective into probably the most universally recognized shape in the world. 


2. Important Circle Theorems Terminology to understand

Before diving into the theorems, let’s spend a moment on some of the key Circle Theorem terms we need to understand. Some of these you may already be familiar with, but no harm in having a refresh!

  • Centre: The exact middle point of a circle.
  • Radius: A straight line extending from the centre to any point on the circle’s edge.
  • Diameter: A line bisecting the circle, passing straight through its centre.
  • Chord: A line segment connecting two points on the circle.
  • Arc: A segment of the circle’s circumference.
  • Sector: A portion of the circle, delineated by two radii and their intercepted arc.
  • Tangent: A line just touching the circle’s edge, without crossing its boundary.


3. Key Circle Theorems:

a) Angles in the Same Segment: Any angle stemming from the same chord or segment within a circle will consistently be equal. It’s a principle rooted in consistent geometry and is one of the key Circle Theormens to understand.

b) The Central and Inscribed Angle Relationship: A central angle is always twice the size of an inscribed angle when both span the same arc. This relationship holds true irrespective of the circle’s size.

c) Tangent-Radius Perpendicularity: Wherever a tangent intersects a circle, and a radius is drawn to that very point of tangency, the two will always meet at a right angle.

d) Opposite Angles of a Cyclic Quadrilateral: For any four-sided figure inscribed in a circle, the angles diagonally opposite one another will invariably sum up to 180°.

e) The Alternate Segment Theorem: When a tangent and triangle intersect within a circle, the angle of intersection between the two will mirror the angle inside the triangle’s opposing vertex.


4. The Relevance of Circle Theorems in Real Life:

Do you really need to know Circle Theorems. Yes! Circle Theorems are on the most widely applicable parts of Geometry in the real world. 

Beyond the classroom, circle theorems lay the foundation for various applications. Architects employ them in building designs, engineers use them to devise mechanical parts, and artists might utilise these principles, perhaps even subconsciously, in their creations.


5. Tips for Navigating Circle Theorems:

  • Diagrammatic representations can often make complex problems much clearer. Simply laying things out in a diagram can really help us see what is going on. 
  • Repetition aids comprehension. The more you engage with these theorems, the more intuitive they become.
  • If confusion arises, revert to basic principles. Often, the solution lies within foundational concepts. Grasp the basic ideas and you will be well set for KS3 Geometry and beyond. 



Circle theorems, while grounded in academic rigour, offer more than abstract mathematical principles. They provide a structured lens through which we can understand and appreciate the circular shapes and patterns integral to our daily lives. So, next time you encounter anything round, take a moment to ponder the rich mathematical tapestry that underpins it. To learn more about Circle Theorems, check our collection of Circle Worksheets here.