One center, one distance — and the most symmetric shape there is, with its first theorem.
Pin one end of a string to the table, hold a pencil at the other end, pull it tight, and swing the pencil all the way around. Every mark it leaves is exactly as far from the pin as every other mark. That single rule — the same distance from one point — is the entire definition of a circle, and it makes the circle the most symmetric figure in all of geometry: spin it, flip it, and it never changes. Out of that symmetry falls our very first circle theorem — a clean perpendicular cut that splits a chord exactly in half — and with the Pythagorean theorem in hand, it already lets us measure a circle we can only partly see.
A circle is the set of all points in a plane that are a fixed distance from a fixed point. The fixed point is the center, written O; the fixed distance is the radius, written r. We name the whole figure ⊙O (read “circle O”) — a circle is known the moment you know its center and its radius.
The compass on your desk is this definition made out of metal: the sharp pivot is the center O, and the opening between the two legs is the radius r. As the pencil leg turns, it cannot get any nearer or any farther from the pivot, so it traces points that are all the same distance from O. Every radius you could draw — from O out to any point on the rim — has that same length r. That one unchanging length is the quiet source of everything else in this stage.
The word radius names two things at once: the segment from the center to a point on the circle, and the length of that segment. “Draw a radius” means the segment; “the radius is 5” means the number. Same word, both meanings — context tells you which.
A few names will carry us through the whole stage. A chord is a segment whose two endpoints lie on the circle. Slide a chord until it passes straight through the center and it becomes as long as a chord can ever be: that longest chord is the diameter. A diameter is two radii laid end to end, so
d = 2r (the diameter is twice the radius).
A piece of the rim itself — the curved part — is an arc. Any two points cut the rim into two arcs: the shorter one is the minor arc and the longer one the major arc. When the two points are the ends of a diameter, the two arcs are equal halves, and each is a semicircle.
A diameter is a chord — it is simply the special longest one, the chord that happens to pass through the center. But the reverse is not true: not every chord is a diameter. A chord earns the name “diameter” only when it runs through O.
Set a circle on a turntable, mark one spot, and spin it. After any turn at all — 7°, 90°, 158.4°, anything — the circle lands exactly back on itself; you could not tell it had moved. No other figure does this for every angle. We say the circle has rotational symmetry about its center through every angle.
It is just as symmetric under folding. Take any line through the center O and fold the page along it: the two halves of the circle land perfectly on each other. Such a fold line is a line of symmetry, and every single line through O — that is, every diameter — works. A square has only 4 lines of symmetry and an equilateral triangle only 3, but a circle has infinitely many lines of symmetry, one for each diameter.
This perfect symmetry is not just pretty — it is why the theorems of this stage come out so clean. Whenever a figure can be folded or spun onto itself, parts that swap places must be equal. Hold that thought; the very next section cashes it in.
Here is the first theorem of the circle, and it comes straight out of the symmetry we just met. Take any chord AB, and from the center O drop a perpendicular down to it, meeting the chord at a point M. The claim is short:
OM ⊥ AB ⇒ AM = MB (the perpendicular from the center bisects the chord).
Why must it be true? Join O to A and O to B. Then OA = OB = r (both are radii), the side OM is shared by both little triangles, and the angles at M are both right angles. So the right triangles △OMA and △OMB are congruent by HL (hypotenuse–leg) — and matching parts of congruent triangles are equal, giving AM = MB. Fold the picture along OM and A lands on B: same fact, seen as symmetry.
Three facts about a line through a chord travel as a team: it passes through the center, it is perpendicular to the chord, it bisects the chord (and it bisects the chord’s arc too). Any two of these three force the third. Know that a line through O meets a chord at right angles, and it must split it in half; know it bisects the chord, and it must run through O.
That perpendicular does more than split the chord — it builds a right triangle, and a right triangle is a thing we know how to solve. Look at △OMA: one leg is OM, the distance from the center to the chord; the other leg is AM, exactly half the chord; and the hypotenuse is OA = r. By the Pythagorean theorem,
r² = OM² + (½ chord)²
This one relation ties together the radius, the chord, and the chord’s distance from the center — give it any two and it hands you the third.
A chord 16 units long sits 6 units from the center. How big is the circle? Half the chord is 8, so
r = √(8² + 6²) = √(64 + 36) = √100 = 10.
Run it the other way: if the radius is 10 and the chord is 16 (half-chord 8), then the chord-center distance is OM = √(10² − 8²) = √(100 − 64) = √36 = 6. The famous 6–8–10 right triangle is hiding inside the circle.
One rule built this whole lesson: a circle is every point a fixed distance r from a center O (write it ⊙O). From that rule:
A circle has radius 5. What is its diameter?
d = 2r = 2 × 5 = 10.
What is the longest chord you can draw in a circle, and what is special about it?
The diameter — it is the chord that passes through the center O, and every other chord is shorter.
A chord 24 long lies 5 units from the center. Find the radius.
The perpendicular from O bisects the chord, so the half-chord is 12. Then r² = OM² + (½ chord)² = 5² + 12² = 25 + 144 = 169, so r = √169 = 13. (The 5–12–13 triangle.)
A circle of radius 10 has a chord 16 long. How far is the chord from the center?
Half-chord = 8, so OM = √(r² − (½ chord)²) = √(10² − 8²) = √(100 − 64) = √36 = 6.
How many lines of symmetry does a circle have, and why?
Infinitely many — one along every diameter. Fold the circle along any line through the center O and the two halves match exactly, so each such line is a line of symmetry, and there are infinitely many lines through O.
The perpendicular from the center meets a chord at M, with AM = 7. How long is the whole chord?
The perpendicular from the center bisects the chord, so MB = AM = 7 and the whole chord AB = 7 + 7 = 14.
Six questions to lock it in. Tap the answer you think is right.
The whole lesson rests on a single definition — a circle is the set of points a fixed distance r from a center — and on its immediate payoff, symmetry. If a learner can say that sentence and point to a radius, a chord, and a diameter, the vocabulary is in place. Then the symmetry (rotational through every angle, reflective across every diameter) is what makes the first theorem feel inevitable rather than memorized: fold along OM and A lands on B.
Two misconceptions are worth heading off directly. First, learners often think “radius” means only a number; reinforce that it names the segment and its length, and likewise for diameter. Second, many believe a diameter is something other than a chord, or that only some diameters are symmetry axes — stress that a diameter is the special longest chord and that every diameter is a line of symmetry. The Pythagorean step (r² = OM² + (½ chord)²) is the same right-triangle reasoning from Stage 15, now living inside a circle.
Common Core: 7.G.B.4 (know the parts and vocabulary of a circle), G-C.A.1 (prove all circles are similar — the seed of why one constant r controls everything), and G-C.A.2 (identify and use relationships among chords and radii; the perpendicular bisector of a chord passes through the center).