Is a diameter also a chord?

Yes. A chord is any segment with both endpoints on the circle, and a diameter satisfies that — it just happens to pass through the center. It is the longest possible chord, and its distance from the center is zero. The reverse is not true: most chords are not diameters, so never assume a chord passes through the center unless the problem says so or marks it.

What is the difference between arc measure and arc length?

Arc measure is in degrees and equals the central angle, so it tells you what fraction of the circle you have; it is the same for a tiny circle and a huge one. Arc length is an actual distance in units, found with s = m/360· 2 r, so it grows with the radius. If your answer needs a unit like inches, you want length; if it needs a degree symbol, you want measure.

When do I name an arc with two letters and when with three?

Use two letters for a minor arc (less than 180^). Use three letters for a major arc (more than 180^) or a semicircle (exactly 180^), with the middle letter naming a point along the path you mean. Three letters remove the ambiguity, since A and B alone could describe either way around the circle.

Why does a perpendicular from the center always bisect a chord?

Draw radii to both endpoints of the chord. Those radii are congruent, so the triangle formed with the chord is isosceles, and the perpendicular from the center is the altitude to the base. In an isosceles triangle the altitude to the base is also the median and the angle bisector, so it splits the chord into two equal pieces and splits the central angle — and therefore the intercepted arc — in half as well.

GEOM-9.1

Circles: Radii, Chords & Arcs

Master circle vocabulary — radius, chord, secant, tangent, arc — plus arc measure vs. arc length and the radius-chord theorems, with worked examples.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Circles: Radii, Chords & Arcs, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Circles: Radii, Chords & Arcs free

What this lesson covers

Almost everything you will do for the rest of this unit rests on the vocabulary and two or three theorems you learn right here. A circle looks simple — one center, one distance — but that single condition (every point on the circle is exactly

r

units from the center) forces a surprising amount of structure: chords get bisected, right triangles appear out of nowhere, and arcs inherit their sizes from angles at the center.

In this lesson you will name every part of a circle precisely, measure arcs in degrees, measure arc length in real units like centimeters, and learn why those two measurements are not the same thing. You will also use the radius-chord relationships to find missing lengths, usually by building a right triangle and applying the Pythagorean Theorem. Get comfortable here and central angles, inscribed angles, and tangent problems in the next lessons will feel like extensions rather than brand-new material.

The Parts of a Circle

A circle is the set of all points in a plane at a fixed distance (the radius) from a fixed point (the center). Circles are named by their centers: circle

O

, written

\odot O

.

The pieces you must be able to identify on sight:

Term	Definition	Key fact
Radius	Segment from center to a point on the circle	All radii of a circle are congruent
Diameter	Chord that passes through the center	$d = 2r$ ; the longest chord
Chord	Segment with both endpoints on the circle	May or may not pass through center
Secant	Line (or ray) that intersects the circle at two points	Contains a chord
Tangent	Line that touches the circle at exactly one point	Perpendicular to the radius at the point of tangency
Arc	Unbroken part of the circle itself	Measured in degrees or in length units

Two distinctions cause most of the early mistakes. First, a chord is a segment while a secant is a line — the secant keeps going past the circle, the chord stops. Second, every diameter is a chord, but most chords are not diameters. Students often write "the chord" when they mean "the diameter" and then wrongly assume it passes through the center.

The point where a tangent touches is the point of tangency. The radius drawn to that point is perpendicular to the tangent line — a fact you will lean on constantly in lesson 9.3, and one that comes from the same idea as the radius-chord theorems below: the shortest distance from the center to a line is the perpendicular distance.

Naming and Measuring Arcs

An arc is a piece of the circle's edge. A central angle is an angle whose vertex is the center; the arc it cuts off is its intercepted arc, and by definition

m\overset{\frown}{AB} = m\angle AOB

where

O

is the center. That equation is the whole definition of arc measure in degrees.

Arcs come in three sizes. A minor arc measures less than

180^\circ

and is named with two letters,

\overset{\frown}{AB}

. A major arc measures more than

180^\circ

and must be named with three letters,

\overset{\frown}{ACB}

, where

C

is a point on the long way around. A semicircle measures exactly

180^\circ

and has a diameter for its endpoints; it also gets three letters. If a problem gives you only two letters, assume the minor arc.

Two rules do nearly all the arithmetic. The Arc Addition Postulate says that if

B

lies on

\overset{\frown}{AC}

, then

m\overset{\frown}{AB} + m\overset{\frown}{BC} = m\overset{\frown}{AC}

. And the arcs of a full circle sum to

360^\circ

, so a minor arc and its major arc satisfy

m\overset{\frown}{AB} + m\overset{\frown}{ACB} = 360^\circ

.

Where students go wrong: writing

\overset{\frown}{AB}

when they mean the major arc, and forgetting that arcs in different circles can have equal measure without being congruent arcs. Congruent arcs must have the same measure and lie in the same circle or in congruent circles. A

60^\circ

arc on a bicycle wheel and a

60^\circ

arc on a dinner plate are not congruent — one is much longer than the other.

Arc Measure Versus Arc Length

This is the single most confused pair of ideas in the unit, and it is worth slowing down for.

Arc measure answers "what fraction of the way around?" It is in degrees and does not depend on the size of the circle. Arc length answers "how far would you walk along the curve?" It is in units — centimeters, feet, miles — and it grows as the circle grows.

Because all circles are similar, the arc of a given degree measure is always the same fraction of the circumference:

s = \frac{m^\circ}{360^\circ}\cdot 2\pi r

So a

90^\circ

arc is one quarter of the way around any circle, but on a circle of radius

4

its length is

\frac{1}{4}(8\pi) = 2\pi

units, while on a radius-

12

circle it is

6\pi

units. Same measure, triple the length.

Question asked	Answer type	Formula
How many degrees is $\overset{\frown}{AB}$ ?	Degrees, $0$ to $360$	$m\overset{\frown}{AB} = m\angle AOB$
How long is $\overset{\frown}{AB}$ ?	Linear units	$s = \frac{m}{360}\cdot 2\pi r$

A good habit: before answering, check whether the problem gives you a radius or diameter. If no length is given anywhere, the answer cannot be an arc length — there is nothing to measure with. If the answer is supposed to include

\pi

or a unit like inches, you are computing length, not measure.

Another frequent slip is using the diameter in place of the radius or forgetting to double the radius. Write the circumference out first, then take the fraction.

Radius-Chord Relationships

Three related theorems come from the fact that a radius drawn to a chord creates isosceles triangles (two sides are radii, hence congruent).

First: if a radius or diameter is perpendicular to a chord, it bisects the chord and its arc. So if

\overline{OM}\perp\overline{AB}

M

, then

AM = MB

and

m\overset{\frown}{AM} = m\overset{\frown}{MB}

.

Second, the converse: the perpendicular bisector of a chord passes through the center. This is how you locate the center of a circular object — draw two chords, construct their perpendicular bisectors, and the intersection is the center.

Third: in the same circle (or congruent circles), two chords are congruent if and only if they are equidistant from the center. Distance from the center always means the perpendicular distance. A consequence: congruent chords cut off congruent arcs, and the closer a chord is to the center, the longer it is — which is why the diameter, at distance

0

, is the longest chord.

In practice, almost every radius-chord problem becomes a right triangle. Draw the radius to an endpoint of the chord (hypotenuse

r

), the perpendicular from the center to the chord (leg

d

), and half the chord (leg

\tfrac{1}{2}c

). Then

r^2 = d^2 + \left(\tfrac{c}{2}\right)^2

The classic error is plugging in the whole chord instead of half of it. The other is using the diameter as the hypotenuse. Label your picture with the half-chord before you touch the Pythagorean Theorem, and both mistakes disappear.

Key terms

Chord.: A segment whose two endpoints both lie on the circle. A diameter is the special chord that passes through the center.
Secant.: A line that intersects a circle at exactly two points; it contains a chord.
Tangent.: A line in the plane of a circle that intersects it at exactly one point, called the point of tangency; it is perpendicular to the radius at that point.
Central angle.: An angle whose vertex is the center of the circle. Its measure equals the measure of its intercepted arc.
Minor arc.: An arc measuring less than 180 degrees, named with its two endpoints, as in arc AB.
Major arc.: An arc measuring more than 180 degrees, named with three letters so the intended path is clear.
Arc measure.: The size of an arc in degrees, equal to its central angle; independent of the circle's radius.
Arc length.: The distance along the arc in linear units, found by $s=\frac{m}{360}\cdot 2\pi r$ .

Worked example

In circle

O

, the radius is 10 cm. Chord

\overline{AB}

lies 6 cm from the center, and

\overline{OM}

is the perpendicular segment from

O

\overline{AB}

at point

M

. (a) Find

AB

. (b) If

m\overset{\frown}{AB} = 74^\circ

, find the length of minor arc

AB

to the nearest hundredth of a centimeter. (c) Find

m\overset{\frown}{AMB}

where the arc is taken the long way around.

(a) Draw radius

\overline{OA}

. Since

\overline{OM}\perp\overline{AB}

, triangle

OMA

is a right triangle with hypotenuse

OA = 10

(a radius) and leg

OM = 6

(the given distance).

Apply the Pythagorean Theorem:

10^2 = 6^2 + AM^2

, so

100 = 36 + AM^2

and

AM^2 = 64

, giving

AM = 8

cm.

Because a radius perpendicular to a chord bisects that chord,

AM = MB

, so

AB = 2(8) = 16

cm. The most common error here is stopping at 8 — that is only half the chord.

(b) Arc length uses the fraction of the circle times the circumference. The circumference is

2\pi r = 2\pi(10) = 20\pi

cm.

s = \frac{74}{360}\cdot 20\pi = \frac{1480\pi}{360} = \frac{37\pi}{9}\approx 12.92\text{ cm}

Notice the answer carries centimeters, because arc length is a distance. The 74 degrees is the measure; the 12.92 cm is the length.

(c) The minor arc and major arc together make the full circle:

m\overset{\frown}{AMB} = 360^\circ - 74^\circ = 286^\circ

. Since this exceeds 180 degrees, the three-letter name is required.

Practice questions

In circle

P

, the diameter is 26 units and chord

\overline{QR}

measures 10 units. How far is

\overline{QR}

from the center

P

5 units
12 units
13 units
24 units

Answer: 12 units

The radius is half the diameter, so

r = 13

. Drop a perpendicular from

P

\overline{QR}

; it bisects the chord, so the half-chord is

5

. The right triangle has hypotenuse 13 and one leg 5, so the distance

d

satisfies

13^2 = 5^2 + d^2

, giving

d^2 = 169 - 25 = 144

and

d = 12

. The choice 5 is the half-chord, and 13 is the radius — both are pieces of the setup, not the answer to the question asked.

Points

A

B

, and

C

lie on circle

O

in that order. If

m\overset{\frown}{AB} = 88^\circ

and

m\overset{\frown}{BC} = 130^\circ

, find

m\overset{\frown}{AC}

for the arc that does not contain

B

, and explain your reasoning.

Answer: $142^\circ$

By the Arc Addition Postulate, the arc from

A

C

that passes through

B

measures

88^\circ + 130^\circ = 218^\circ

. The three arcs

\overset{\frown}{AB}

\overset{\frown}{BC}

, and the remaining arc

\overset{\frown}{AC}

together make the entire circle, which is

360^\circ

. So the arc not containing

B

measures

360^\circ - 218^\circ = 142^\circ

. Since it is less than

180^\circ

, it is a minor arc and the two-letter name

\overset{\frown}{AC}

is appropriate.

Circle

M

has radius 3 inches and circle

N

has radius 9 inches. Each circle contains a

40^\circ

arc. Compare the arc measures and the arc lengths, showing your computations.

Answer: Both arc measures are $40^\circ$ ; the arc in circle $M$ is $\frac{2\pi}{3}$ inches long and the arc in circle $N$ is $2\pi$ inches long, three times as long.

Arc measure depends only on the central angle, so both arcs measure

40^\circ

— this is why arcs of equal measure in different circles are not automatically congruent. For length, circle

M

gives

s = \frac{40}{360}\cdot 2\pi(3) = \frac{1}{9}\cdot 6\pi = \frac{2\pi}{3}\approx 2.09

inches, and circle

N

gives

s = \frac{1}{9}\cdot 18\pi = 2\pi \approx 6.28

inches. Tripling the radius triples the arc length while leaving the degree measure unchanged, which is the clearest illustration of the difference between the two quantities.

FAQ

Is a diameter also a chord?: Yes. A chord is any segment with both endpoints on the circle, and a diameter satisfies that — it just happens to pass through the center. It is the longest possible chord, and its distance from the center is zero. The reverse is not true: most chords are not diameters, so never assume a chord passes through the center unless the problem says so or marks it.
What is the difference between arc measure and arc length?: Arc measure is in degrees and equals the central angle, so it tells you what fraction of the circle you have; it is the same for a tiny circle and a huge one. Arc length is an actual distance in units, found with $s = \frac{m}{360}\cdot 2\pi r$ , so it grows with the radius. If your answer needs a unit like inches, you want length; if it needs a degree symbol, you want measure.
When do I name an arc with two letters and when with three?: Use two letters for a minor arc (less than $180^\circ$ ). Use three letters for a major arc (more than $180^\circ$ ) or a semicircle (exactly $180^\circ$ ), with the middle letter naming a point along the path you mean. Three letters remove the ambiguity, since $A$ and $B$ alone could describe either way around the circle.
Why does a perpendicular from the center always bisect a chord?: Draw radii to both endpoints of the chord. Those radii are congruent, so the triangle formed with the chord is isosceles, and the perpendicular from the center is the altitude to the base. In an isosceles triangle the altitude to the base is also the median and the angle bisector, so it splits the chord into two equal pieces and splits the central angle — and therefore the intercepted arc — in half as well.

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