Circle Geometry
Understanding Circle Geometry
Basic Definitions
- A circle is a two-dimensional shape made by drawing a curve that is always the same distance from a centre point.
- The radius of a circle connects the centre of the circle with a point on the circle.
- The diameter of a circle is twice the length of the radius, connecting two points on the circle passing through the centre.
- The circumference of a circle is the distance around it and it can be calculated using the formula Circumference = 2πr (where r is radius) or πd (where d is diameter).
- A chord of a circle is a line segment that connects any two points on the circle.
- The arc of a circle is any part of the circumference of the circle.
- A sector is the area of a circle enclosed by two radii and the corresponding arc.
- A segment in a circle is the region between a chord and either of the two arcs defined by the chord.
Key Theorems and Properties
- All radii of a circle are equal in length.
- The line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.
- Angle at the centre of a circle is twice the angle at the circumference, from the same two points on the circle.
- Angles in the same segment of a circle are equal.
- The angle subtended by the diameter at the circumference is a right angle.
- The opposite angles of a cyclic quadrilateral (four-sided figure inscribed in a circle) always add to 180 degrees.
Area of a Circle and Sectors
- The area of a circle can be calculated by the formula Area = πr², where r is the radius.
- The area of a sector can be calculated by finding the fraction of the full circle that the sector represents, and then multiplying this by the area of the full circle. If θ is the angle at the centre of the sector, then Area of sector = (θ/360) * πr².
It’s important to know these definitions, theorems, and formulas, and to practice applying them to problem-solving tasks. Understanding circle geometry can form a solid foundation for studying more complex mathematical concepts.