Cyclic Quadrilaterals
Opposite angles of a quadrilateral inscribed in a circle sum to 180°.
Key idea
Opposite angles of a quadrilateral inscribed in a circle sum to 180°.
In depth
A quadrilateral with all four vertices on a circle has opposite angles adding to 180°. The converse also holds, which is a way to prove four points lie on one circle.
Worked example
If one angle is 80°, the opposite angle is 100°.
Real-world uses
- Designing a four-sided window frame that fits a circle
- Checking circular garden layouts
Common mistake
Applying it to quadrilaterals that are not inscribed.
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 Cyclic Quadrilaterals?
Opposite angles of a quadrilateral inscribed in a circle sum to 180°. It is a topic at the expert level in Geometry.
Where is Cyclic Quadrilaterals used in real life?
Real-world examples: Designing a four-sided window frame that fits a circle; Checking circular garden layouts.
What should I learn before Cyclic Quadrilaterals?
Make sure you are comfortable with Power of a Point, the previous topic in Geometry.
What is a common mistake with Cyclic Quadrilaterals?
Applying it to quadrilaterals that are not inscribed.
How can I practise Cyclic Quadrilaterals?
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 Cyclic Quadrilaterals?
Move on to Volume of Prisms and Cylinders next.
Related expert topics in Geometry
- Power of a Point – For a point and a circle, products of segment lengths along secants are equal.
- Cavalieri’s Principle – Solids with equal cross-sectional areas at every height have equal volume.
- Area by Coordinates (Shoelace) – Area of a polygon from vertices.
- Parabola (Geometric Definition) – Points equidistant from a focus and a directrix.
- Ellipse – Sum of distances to two foci is constant.
- Hyperbola – Difference of distances to two foci is constant.