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 = ˝ 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 OE˛ = AO˛ - AE˛
Segment Height ED = Radius AO - Apothem OE

2) Radius AO & Chord AB
AE = ˝AB
From the Pythagorean Theorem OE˛ = AO˛ - AE˛
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) 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 = ˝AB then:
CE= (˝AB •˝AB) / ED
CE = AB˛ / 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

6) Chord AB & Apothem OE
AE = ˝AB
From the Pythagorean Theorem
Radius AO˛ = OE˛ + AE˛
Segment ED = Radius AO - Apothem OE
Angle AOE = arc tangent (AE/OE)
Central Angle AOB = 2 • Angle AOE

7) 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
AE˛ = AO˛ - OE˛
Chord AB = 2 • AE

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