Math101 · Independent Ontario learning library
Math101learn.math101.caCircles
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.
Worked example
Representations and interpretation
A centre-radius diagram generates every point by rotating a radius through a full turn. A coordinate equation represents the same fixed-distance locus, while sector and chord diagrams isolate local pieces of the circle.
Reasoning about variations
Scaling radius by $k$ scales every chord and circumference by $k$ but every area by $k^2$. Concentric circles form an annulus whose width is the difference of radii, not the difference of areas.
Common mistakes
How to check your work
- Attach linear or square units appropriately.
- Compare a result with diameter and full circumference bounds.
- Use symmetry or the coordinate distance to verify claimed points.
Practice
- A circle has circumference $20\pi$ cm. Find its radius.
- If radius doubles, how does circumference change?
- What set defines a circle?
Answers and brief solutions
Show answers
- $10$ cm $2\pi r=20\pi$ gives $r=10$.
- It doubles $C=2\pi r$ is linear in radius.
- All plane points at one fixed distance from a centre That fixed-distance condition is the locus definition.
Synthesis and transfer
Drawing a circle with a compass enforces the locus definition physically; measuring several radii checks constancy while a noncentral chord demonstrates what is not a diameter.
Fixing one compass point at $O$ and keeping the opening at $r$ generates only points whose distance from $O$ is $r$. A point closer than $r$ lies inside the disk but not on the circle; a point farther away is outside. This distance definition remains valid in any orientation and becomes $(x-h)^2+(y-k)^2=r^2$ in coordinates. Scaling all distances from the centre produces a concentric circle, while translating the centre moves the locus without changing radius. Distinguishing the one-dimensional boundary from the filled two-dimensional disk prevents area and circumference language from being interchanged.
Related topics
Teaching and accessibility note
Explore the idea
Geometry measurement
Change one quantity at a time and connect what moves to Circles.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
A circle has circumference $20\pi$ cm. Find its radius.
- $2\pi r=20\pi$ gives $r=10$.
End of lesson
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