Foundations — Circles — centre, radius, diameter, chord, arc, sector, segment

Before you can use the parent note, you need to be able to read it. Below is every symbol and idea it silently assumes, built from zero, each one anchored to a picture. We go slowly on purpose — nothing here is used before it is drawn.

1. A point, a distance, and the birth of the circle

Look at the first figure. Start with one dot and call it $O$. Now imagine a piece of string of fixed length pinned to $O$. Swing the free end all the way around: the pencil traces a perfectly round curve. That curve is the circle, and the pinned dot is its centre.

• The pin is the centre, written $O$.
• The taut string length is the radius, written $r$.
• Every point the pencil visits is exactly $r$ away from $O$ — that "exactly, always the same" is the whole definition.

2. Symbols $O$, $r$, $P$ — naming things so we can talk about them

Mathematics uses single letters as short nicknames. You must know each nickname before it appears in a formula.

So the sentence "$OP=r$" reads out loud as: the distance from the centre to the point P equals the radius. That is the circle's rule written in symbols. Nothing more mysterious than that.

3. Coordinates and the symbol $(x,y)$ — with signs

The parent note suddenly writes points as $(x,y)$ and uses $\sqrt{x^2+y^2}$. Here is where that comes from.

Why the topic needs these. To find the distance from the centre $O=(0,0)$ to a point $(x,y)$, the picture shows a right-angled triangle: the horizontal leg has length $|x|$, the vertical leg has length $|y|$. To turn those two legs into the straight-line distance we need the Pythagorean theorem:

${\text{distance}}^2=x^2+y^2⟹\text{distance}=\sqrt{x^2+y^2}$

Because of the squares, the signs wash out and a point at $(-3,4)$ is just as far ($5$) as one at $(3,4)$. Setting that distance equal to $r$ gives the circle's rule in coordinates:

$\sqrt{x^2+y^2}=r⟹x^2+y^2=r^2$

4. Chord, and the symbol $d$ for diameter

The parent talks about "chords" from the start — here is what one actually is, before we ever slice the circle.

The diameter derivation quietly uses "if three points are on a straight line, the whole equals the sum of the parts." That deserves a picture.

Because both $AO$ and $OB$ are radii ($=r$), the diameter $d=AB=r+r=2r$. This is the only reason $d=2r$ is true — it is addition, nothing deeper.

5. Arc and circumference — the curved edge and how far around

6. Angles and the symbol $\theta$ — with signs

Arc, sector and segment all depend on an angle at the centre. You must know what $\theta$ means, in what units, and which way is positive.

There are two ways to measure turning:

Stacking that definition up gives the clean rule the parent uses:

$s=r\theta (\theta \text{ in radians}).$

Convert between them by matching full turns: $360^{\circ }=2\pi$ radians, so

${\theta }_{\text{rad}}={\theta }_{\text{deg}}\times \frac{\pi }{180}.$

7. The symbol $\pi$

8. Sector and segment — the two ways to cut a region

The opening promised these words; here they finally get a picture. Note both are regions (filled areas), not just lines.

9. $\sin \theta$ — the height-maker, for every angle

The segment formula uses $\sin \theta$. Here is the fact about it you need, across the full range of centre angles.

That is why the triangle inside a sector has area $\frac{1}{2}\cdot r\cdot r\sin \theta =\frac{1}{2}{r}^2\sin \theta$ (area $=\frac{1}{2}\times \text{base}\times \text{height}$), valid for the whole sector range.

10. The integral symbol $\int$ (a peek, not a demand)

The parent uses ${\int }_0^{r}2\pi rdr$ to prove area $=\pi r^2$.

How the foundations feed the topic

Read it top-down: the two seeds — a point and a distance — grow the whole family of circle facts. Angles branch off into arcs and sectors; sine finishes the segment.

Equipment checklist

Cover the right side and test yourself. If any answer is fuzzy, re-read that section above.

Parent: Circles — centre, radius, diameter, chord, arc, sector, segment · Next tools: Angle Properties in Circles, Circle Theorems, Sectors and Pizza Problem.