Segment areas

Objective. Combine sectors and triangles to find segment areas.

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Learn: the key idea

Segment areas are found by subtracting the area of the triangle formed by the two radii and chord from the area of the whole sector that contains it. First calculate the sector area using (θ/360) × πr², then calculate the triangle area using (1/2)r² sin θ, and subtract. For example, with r = 5 cm and θ = 90°, sector area = (1/4)π(25) and triangle area = (1/2)(25)(1) = 12.5, giving the segment as the difference. This only works directly for the minor segment; for the major segment, subtract the minor segment from the full circle area instead.

Worked example

A circle has radius 8 cm. A sector has angle 60° at the centre. Find the area of the minor segment, to 1 decimal place.

1 $Sector area = (60/360) \times π(8²) = (1/6) \times 64π \approx 33.51 cm²$
2 $Triangle area = (1/2)(8²) sin 60° = 32 \times 0.8660 \approx 27.71 cm²$
3 $Segment area = sector - triangle = 33.51 - 27.71$
4 $Segment area \approx 5.8 cm²$

Answer: 5.8 cm²

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Common mistake. Students sometimes subtract in the wrong order, computing triangle area minus sector area, giving a negative result; always take the larger sector area minus the triangle area for a minor segment.

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