Math101 · Independent Ontario learning library
Math101Circles
A circle is the locus of all points in a plane at fixed distance $r>0$ from a centre. Its symmetry makes circumference and area depend only on radius.
Circles model rotation, wheels, waves, optics, design, and periodic motion. The locus definition unifies synthetic geometry and coordinate equations.
Intuition and core definition
A circle is the locus of all points in a plane at fixed distance $r>0$ from a centre. Its symmetry makes circumference and area depend only on radius. Circle geometry connects chords, arcs, central and inscribed angles, tangents, and coordinate distance.
Notation, language, and conditions
Radius $r$, diameter $d=2r$, circumference $C=2\pi r=\pi d$, and area $A=\pi r^2$ describe distinct quantities and units. Congruent circles have equal radii; concentric circles share a centre. An arc’s degree measure equals its central angle measure.
Why this idea matters
A circle is the locus of points at one fixed distance from a centre, unifying its geometric construction and coordinate equation.
A dependable method
- Identify centre, radius, and whether the requested quantity is length, area, angle, or locus.
- Choose a theorem whose objects and conditions match the diagram.
- Translate diameters to radii and degrees to turn fractions when needed.
- Calculate with exact $\pi$ until approximation is requested.
- Check dimensions, bounds, and symmetry.
