A chord
Pick two points on a circle. The straight line joining them is called a chord. A chord that passes through the center is a diameter, and no chord is longer than a diameter.
A chord: a straight line from one point on the circle to another.
Two arcs
An arc is a part of the circle itself, the curved edge between two points on it. The same two points split the circle into two arcs, one each way round. The shorter one is the minor arc and the longer one is the major arc. Together they make the whole circumference.
The chord and the minor arc join the same two points, but the chord goes straight across and the arc follows the curve, so the chord is always the shorter of the two.
When the two points are the ends of a diameter, the two arcs are equal, and each is half the circle, a semicircle.
The minor arc between the two points: the shorter way round.
The major arc between the same two points: the longer way round.
A sector: two radii and an arc
Join the two points to the center instead of to each other. The two radii and the arc between their ends close off a region called a sector, shaped like a slice of a round pizza. Its corner is at the center.
The two radii cut the circle into two sectors. The one with the minor arc is the minor sector, and the one with the major arc is the major sector. A quarter circle is a sector whose radii are at right angles, and a half circle is a sector whose radii make a straight line.
A sector: the region closed off by two radii and the arc between them.
A segment: a chord and an arc
Now cut along the chord. The chord and the arc beside it close off a region called a segment. A segment has no corner at the center: it stops at the chord.
The chord cuts the circle into two segments. The one with the minor arc is the minor segment, and it does not contain the center. The one with the major arc is the major segment, and the center is inside it. A diameter cuts the circle into two equal segments, the two semicircles.
The minor segment: the region between the chord and the minor arc. The center is outside it.
The major segment on the same chord: the region between the chord and the major arc, with the center inside it.
Telling a sector from a segment
Both regions have an arc for one edge. What closes them off is different. A sector is closed by two radii, which meet at the center. A segment is closed by one chord.
On the same two points, the minor sector is the minor segment together with the triangle made by the two radii and the chord. So the area of the minor segment is the area of the sector minus the area of that triangle.
The two radii and the chord make a triangle with its corner at the center. The sector is that triangle and the shaded segment together.
The area of a segment, for a quarter circle
When the angle between the two radii is 90°, the sector is a quarter circle, and the triangle has a right angle at the center, so its base and its height are both radii.
Take a radius of 6 cm, and use 3.14 for . The whole circle has an area of 3.14 × 6 × 6 = 113.04 cm², so the quarter circle has an area of 113.04 ÷ 4 = 28.26 cm². The triangle has an area of ½ × 6 × 6 = 18 cm². The minor segment has an area of 28.26 − 18 = 10.26 cm².
The major segment is everything else: 113.04 − 10.26 = 102.78 cm², which is the other three quarters of the circle, 84.78 cm², together with the triangle, 18 cm².
The usual mistakes
Calling a segment a sector. If the region reaches the center along two radii, it is a sector; if it stops at a chord, it is a segment.
Mixing up a chord and an arc. The chord is the straight line across; the arc is the curved part of the circle between the same two points.
Calling part of the circumference the circumference. The circumference is the whole way round; an arc is only part of it.
Worked example: The Concrete Below the Road in a Round Tunnel
Question A road tunnel has a circular cross-section of radius 4 m and center O. The flat road surface runs from A to B across the circle, and angle AOB is 90°. The part of the circle below the road is filled with concrete. Take π = 3.14. (a) Find the area of the concrete in the cross-section. (b) Find the area of the cross-section that is left open above the road.
1.The concrete is the minor segment cut off by the chord AB. Its area is the area of sector AOB minus the area of triangle AOB.
The concrete fills the segment below the chord AB: the sector AOB with the triangle AOB taken away. 2.Angle AOB is 90°, a quarter turn, so the sector is a quarter of the circle: 90360 × π × 42 = 14 × 3.14 × 16 = 12.56 m2.
Angle AOB is 90°, a quarter turn: the sector is 14 × 3.14 × 42 = 12.56 m2. 3.OA and OB are at right angles, so triangle AOB has base 4 m and height 4 m: its area is 12 × 4 × 4 = 8 m2.
OA and OB are at right angles, so triangle AOB is 12 × 4 × 4 = 8 m2. 4.(a) The concrete has an area of 12.56 − 8 = 4.56 m2.
(a) The concrete is 12.56 − 8 = 4.56 m2. 5.The whole cross-section is π × 42 = 3.14 × 16 = 50.24 m2. (b) The open part is 50.24 − 4.56 = 45.68 m2. Check: the open part is three quarters of the circle plus the triangle, 37.68 + 8 = 45.68 m2.
(b) The whole circle is 3.14 × 16 = 50.24 m2, so the open part is 50.24 − 4.56 = 45.68 m2.
Answer: (a) 4.56 m2; (b) 45.68 m2
Common mistakes
- Taking the concrete to be the whole sector AOB, 12.56 m2. The sector reaches up to the center O, which is above the road; the concrete stops at the chord, so the triangle AOB must be taken away.
- Using 12 × 4 × 4 for the triangle without checking the angle. The base and height of the triangle are the two radii only because the angle between them is 90°.
More congruence, similarity and circle theorems problems, worked step by step →