Between the arc and the chord

The idea

A segment of a circle is the region bounded by an arc of the circle and the chord joining the endpoints of the arc.

A chord cuts the circle

A chord is a straight line joining two points A and B on a circle. It cuts the circle into two parts. You have already met the sector, which uses the centre and two radii.

Draw the chord AB and shade the region between the arc AB and the chord AB. Is this the same region as the sector?

• Yes, the same
• No, it is smaller
• No, it is bigger

Sector and segment

Show the sector (dashed outline) and the segment (orange). Try two different angles and see how they differ.

An arc and a chord

A segment of a circle is the region bounded by an arc of the circle and the chord joining the endpoints of the arc.

A sector is bounded by an arc and two radii. A segment is bounded by an arc and one chord. The chord is a straight line across the circle, so the segment does not reach the centre unless the chord is a diameter.

Notes

A segment of a circle is the region bounded by an arc of the circle and the chord joining the endpoints of the arc.

Check yourself

What bounds a segment of a circle?

How many straight edges does the border of a segment have?

Answer: 1

The arc is curved and the chord is straight, so a segment has 1 straight edge. A sector has 2.

A chord AB is a diameter of the circle. What are the two segments?

The sector OAB and the segment on the arc AB share the same arc. How do the two regions differ?

• An arc and the two radii to its endpoints. That bounds a sector. A segment uses a chord instead of the two radii.
• Two arcs. A segment has one arc and one straight chord.
• An arc and the chord joining its endpoints — correct. Yes. The chord closes up the region under the arc.
• Two equal semicircular discs — correct. Yes. A diameter cuts the circle into two equal halves, and each half is a segment.
• One segment only. The chord cuts the circle into two parts, and each part is a segment.
• Two sectors with 90° each. They are halves, not quarters, and the border is an arc with a chord, not two radii.
• They are the same region with two names. They are different regions. The sector reaches the centre O and the segment does not (unless AB is a diameter).
• The sector is bounded by two radii, the segment by the chord AB — correct. Yes. Same arc, but the sector closes up through the centre O and the segment closes up with the chord.
• The segment has the arc and the sector does not. Both have the same arc AB. They differ in how they close up.