Parts of a Circle and Formulas


Here are the procedures by which the Circle Calculator determines all of a circle's data from just 2 variables.

Intersecting Chord Theorem
A great time-saver for these calculations is a little-known geometric theorem which states that whenever 2 chords (in this case AB and CD) of a circle intersect at a point E, then AE • EB = CE • ED
Yes, it turns out that "chord" CD is also the circle's diameter and the 2 chords meet at right angles but neither is required for the theorem to hold true.

1) Radius and Central Angle
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.

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