Same distance, same length — and further out is shorter
Two chords sit at different distances from a centre. Which one is longer?
- A circle and its centre O.
- Chord one, with its distance from O marked.
- Chord two, the same distance from O.
- Both chords, measured end to end, come out equal.
The rule
In one circle, the chord nearer the centre is longer. Distance out means the length of the 90° line from the centre.
Half the chord, the distance and the radius form a right triangle. The radius is its longest side. Square the radius. Take away the square of the distance from the centre. What is left gives the half-length squared. The half-length is half the chord. So equal distances give equal half-lengths.
- In one circle, equal distance out gives equal length. The chords may point any way.
- In one circle, further out always means a shorter chord.
- In one circle, equal chords make equal angles at the centre. Each such angle lies between the two radii to the chord's ends.
A tray, measured
The tray's radius is 25 cm. Four strips are laid across it. Take the square root of column three for the half-length. Double it for the strip.
| Strip | Distance out | 25² minus that squared | Strip length |
|---|---|---|---|
| A | 7 cm | 625 − 49 = 576 | 48 cm |
| B | 15 cm | 625 − 225 = 400 | 40 cm |
| C | 20 cm | 625 − 400 = 225 | 30 cm |
| D | 24 cm | 625 − 576 = 49 | 14 cm |
A diameter fits this rule too. Set the distance to 0. Nothing is taken away, so the most is left. On this tray the half is 25 cm. The chord is 50 cm. No chord is longer.
In the lab, move chord B. Watch when it matches chord A.
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