We know the central angle is AOB and so angle AOE = ½ central angle

From trigonometry we know that the sine of angle AOE = AE/AO

So, line AE = sine of angle AOE • line AO

Using the Pythagorean Theorem line OE² = AO² - AE²

Segment Height ED = Radius AO - Apothem OE

2) Radius AO & Chord AB

AE = ½AB

From the Pythagorean Theorem
OE² = AO² - AE²

Segment Height ED = Radius AO - Apothem OE

Angle AOE = arc tangent (AE/OE)

Central Angle AOB = 2 • Angle AOE

3) Radius AO & Segment Height ED

Apothem OE = Radius AO - Segment Height ED

Angle AOE = arc tangent (AE/OE)

Central Angle AOB = 2 • Angle AOE

4) Radius AO & Apothem OE

Segment Height ED = Radius AO -Apothem OE

Angle AOE = arc tangent (AE/OE)

Central Angle AOB = 2 • Angle AOE

5) Radius AO & Arc AB

Circumference = 2 • π • radius AO

Central Angle AOB = (Arc AB / Circumference) • 360

Angle AOE = Central Angle AOB / 2

Chord AB = 2 • sine (Angle AOE) • radius

6) Chord AB & Segment Height ED

This is where the "intersecting chord theorem" *really* comes in handy.

CE • ED = AE • EB

CE = (AE • EB) / ED

Since AE = EB = ½AB then:

CE= (½AB • ½AB) / ED

CE = AB² / 4•ED

Radius AO = (CE + ED) / 2

Apothem OE = Radius AO - Segment Height ED

Angle AOE = arc tangent (AE/OE)

Central Angle AOB = 2 • Angle AOE

7) Chord AB & Apothem OE

AE = ½AB

From the Pythagorean Theorem

Radius AO² = OE² + AE²

Segment ED = Radius AO - Apothem OE

Angle AOE = arc tangent (AE/OE)

Central Angle AOB = 2 • Angle AOE

8) Segment Height ED & Apothem OE

Radius AO = Segment Height ED + Apothem OE

Angle AOE = arc tangent (AE/OE)

Central Angle AOB = 2 • Angle AOE

From the Pythagorean Theorem

AE² = AO² - OE²

Chord AB = 2 • AE

9) Chord AB & Arc Length AB (curved blue line)

There is *no formula* that can solve for the other parts of a circle if you only know the chord and the arc length.

There is a procedure called Newton's Method which can produce an answer. To try it, click the link here and then scroll about ¾ of the way down to "A Real World Example" where we have a worked out example.