Sector minus triangle

The idea

Area of a segment = area of the corresponding sector – area of the corresponding triangle, that is, θ/360 × πr² – area of Δ OAB.

A sector with a triangle inside

In a circle of radius 14 cm, the angle AOB is 90°. The sector OAB has area 154 cm², and the triangle OAB has area 98 cm² (π = 22/7). The blue curved piece between the chord AB and the arc is the segment.

What is the area of the segment?

• 252 cm²
• 56 cm²
• 154 cm²

Pull the triangle out of the sector

Drag the grey triangle down, or press the button. What is left of the sector is the blue segment. Change the angle with the dial and do it again.

Take away the triangle

Area of a segment = θ/360 × πr² − area of Δ OAB, that is, area of the sector minus area of the triangle.

The sector is made of the triangle OAB and the segment, so the segment is what is left when the triangle is taken away. With r = 14 cm and θ = 90°: 154 − 98 = 56 cm².

To find the area of Δ OAB when θ is not 90°, draw OM perpendicular to AB. M is the mid-point of AB and the angle AOM is half of θ. For θ = 120°, OM = r cos 60° and AM = r sin 60°, and the area is ½ × AB × OM.

Notes

Area of a segment = area of the sector − area of Δ OAB = θ/360 × πr² − area of Δ OAB.

Check yourself

A chord of a circle of radius 10 cm makes a right angle at the centre. Find the area of the minor segment. (Take π = 3.14)

Answer: 28.5 cm²

Sector = 90/360 × 3.14 × 100 = 78.5. Triangle = ½ × 10 × 10 = 50. Segment = 78.5 − 50 = 28.5 cm².

A chord of a circle of radius 7 cm subtends a right angle at the centre. Find the area of the minor segment. (Take π = 22/7)

Answer: 14 cm²

Sector = 154 / 4 = 38.5. Triangle = ½ × 7 × 7 = 24.5. Segment = 38.5 − 24.5 = 14 cm².

The radius of a circle is 21 cm and the angle AOB is 120°. The sector has area 462 cm² and the triangle OAB has area 441√3 / 4 cm². What is the area of the segment AYB?

Why is the area of the minor segment always less than the area of its sector?

• (462 − 441√3 / 4) cm² — correct. Yes! Segment = sector − triangle.
• (462 + 441√3 / 4) cm². That adds the triangle. The segment is the sector with the triangle taken away.
• (441√3 / 4 − 462) cm². That is the wrong way round, and it is negative. The sector (462) is bigger than the triangle (about 191).
• The sector is the segment plus the triangle OAB. — correct. Yes! Sector = segment + Δ OAB, so the segment is smaller by the area of the triangle.
• The arc of the segment is shorter than its chord.. An arc is always longer than its chord. This is not the reason.
• The segment has a smaller angle at O.. A segment has no angle at O. It is the piece between a chord and an arc.