To find the area of a major segment when you know the smaller central angle, calculate the minor segment first. Subtract the triangle area from the minor sector area, then subtract the resulting minor segment from the whole circle.
For a circle of radius 6 cm and a chord subtending a smaller angle of 120°, the minor sector has area 12π cm² and the triangle has area 9√3 cm². The major segment is 24π + 9√3 cm². The 2027 SEC G3 Mathematics K310 syllabus covers sector and segment areas.
- Identify the larger region between the chord and the arc.
- Find the minor sector and its triangle using the smaller angle.
- Subtract the minor segment from the area of the circle.
- Keep area units and round only the final result.
CHAPTER 1 OF 4 · Back to contents
Mark the chord and the two segments
Draw a circle with centre O and a chord AB. A chord joins two points on the circumference and divides the circle into two segments. The major segment follows the longer arc from A to B and contains O in a non-diameter example.
Join O to A and B to draw two radii. The smaller central angle lies inside triangle OAB. The minor sector follows the shorter arc and contains the triangle plus the small curved region beyond the chord. This curved region is the minor segment.
Shade the requested region before writing the first calculation. If you shade a region bordered by two radii, you have picked a sector. A segment is bounded by a chord and an arc.
CHAPTER 2 OF 4 · Back to contents
Work the example with a 120° smaller angle
Use radius 6 cm. A 120° sector is one third of a full turn, so the minor sector area is (120 ÷ 360) × π × 6² = 12π cm². The triangle has two sides of 6 cm enclosing the 120° angle, so its area is ½ × 6 × 6 × sin 120° = 9√3 cm².
The minor segment area is 12π − 9√3 cm². The whole circle has area 36π cm², so the major segment has area 36π − (12π − 9√3) = 24π + 9√3 cm², about 91.0 cm² to three significant figures.
Check the sign of the triangle term by imagining the major sector. Its area is 24π cm². The major segment contains that sector and triangle OAB, so its area must be greater than 24π. This check catches a sign error before you round.
Contents · Previous chapter · Next chapter
CHAPTER 3 OF 4 · Back to contents
Practise with radians and check the boundary
Try a circle of radius 8 m with a chord subtending a smaller angle of π/2 radians. The minor sector area is ½ × 8² × π/2 = 16π m². The triangle has area 32 m², since the two radii meet at right angles. The major segment area is 48π + 32 m².
When the smaller angle is in radians, the sector formula ½r²θ uses that angle directly. If the question gives degrees, either convert to radians first or use the fraction of 360° method. Set the calculator to the corresponding angle mode when evaluating sine.
For a question asking for the perimeter of the same major segment, follow the boundary. It consists of the major arc and chord AB. In the 8 m example, the major angle is 3π/2 and the chord is 8√2 m by Pythagoras, giving a perimeter of 12π + 8√2 m.
Contents · Previous chapter · Next chapter
CHAPTER 4 OF 4 · Back to contents
Frequently asked questions
Why does the triangle get added in the final expression? The minor segment excludes the triangle. Subtracting the minor segment from the circle leaves the triangle on the major side of the chord.
What if the chord passes through the centre? It is a diameter and both segments are semicircles. The two areas are equal, so there is no larger segment.
What if the reflex angle is the one printed? Subtract it from a full turn to find the smaller angle for the triangle and minor sector.
What is a quick check at home? Ask the student to show the boundary of the requested region and predict whether its area should be greater or smaller than half the circle.
Contents · Previous chapter · Find a sector perimeter from its arc
Official syllabus and next reading
This independent study guide uses original practice examples. See the official 2027 SEC G3 K310 syllabus and the SEAB syllabus list for G3 school candidates.