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Circular Segment — Definition, Formula & Examples

A circular segment is the region of a circle enclosed between a chord and the arc that the chord cuts off. It looks like a "slice" that has been trimmed by a straight cut across the circle.

Given a circle of radius rr and a chord that subtends a central angle θ\theta (in radians), the circular segment is the region bounded by the chord and the minor (or major) arc. Its area equals the area of the corresponding circular sector minus the area of the triangle formed by the two radii and the chord.

Key Formula

A=12 r2 (θ−sin⁡θ)A = \frac{1}{2}\,r^2\,(\theta - \sin\theta)
Where:
  • AA = Area of the circular segment
  • rr = Radius of the circle
  • θ\theta = Central angle subtended by the chord, in radians

How It Works

To find the area of a circular segment, start by identifying the central angle θ\theta that corresponds to the chord. Compute the area of the circular sector (the "pie slice" from the center) using 12r2θ\frac{1}{2}r^2\theta. Then subtract the area of the isosceles triangle formed by the two radii and the chord, which is 12r2sin⁡θ\frac{1}{2}r^2\sin\theta. The difference gives the area of the segment. If θ\theta is greater than π\pi radians (180°), you get the major segment; if less, the minor segment.

Worked Example

Problem: A circle has radius 10 cm. A chord subtends a central angle of 90° (π/2 radians). Find the area of the minor segment.
Step 1: Convert the angle to radians if needed. Here θ = 90° = π/2 radians.
θ=π2\theta = \frac{\pi}{2}
Step 2: Compute the sector area using (1/2)r²θ.
Asector=12(10)2⋅π2=100π4=25π≈78.54 cm2A_{\text{sector}} = \frac{1}{2}(10)^2 \cdot \frac{\pi}{2} = \frac{100\pi}{4} = 25\pi \approx 78.54 \text{ cm}^2
Step 3: Compute the triangle area using (1/2)r² sin θ.
Atriangle=12(10)2sin⁡ ⁣(π2)=50⋅1=50 cm2A_{\text{triangle}} = \frac{1}{2}(10)^2 \sin\!\left(\frac{\pi}{2}\right) = 50 \cdot 1 = 50 \text{ cm}^2
Step 4: Subtract the triangle area from the sector area to get the segment area.
A=25π−50≈78.54−50=28.54 cm2A = 25\pi - 50 \approx 78.54 - 50 = 28.54 \text{ cm}^2
Answer: The area of the minor segment is approximately 28.54 cm².

Another Example

Problem: A circle has radius 6 m. A chord subtends a central angle of 120° (2π/3 radians). Find the area of the minor segment.
Step 1: Convert to radians: θ = 120° = 2π/3.
θ=2π3\theta = \frac{2\pi}{3}
Step 2: Apply the segment area formula directly.
A=12(6)2 ⁣(2π3−sin⁡2π3)=18 ⁣(2π3−32)A = \frac{1}{2}(6)^2\!\left(\frac{2\pi}{3} - \sin\frac{2\pi}{3}\right) = 18\!\left(\frac{2\pi}{3} - \frac{\sqrt{3}}{2}\right)
Step 3: Evaluate numerically.
A=18(2.0944−0.8660)=18(1.2284)≈22.11 m2A = 18(2.0944 - 0.8660) = 18(1.2284) \approx 22.11 \text{ m}^2
Answer: The area of the minor segment is approximately 22.11 m².

Why It Matters

Circular segments appear in geometry courses whenever you need the area of a region cut off by a chord — for example, finding the cross-sectional area of water in a partially filled pipe. Civil engineers calculate segment areas to design arched windows, tunnel cross-sections, and curved road boundaries. Mastering this concept also strengthens your ability to combine sector and triangle formulas, a skill tested frequently on standardized exams like the SAT and ACT.

Common Mistakes

Mistake: Using degrees directly in the formula A = ½r²(θ − sin θ).
Correction: The formula requires θ in radians. Convert degrees to radians first by multiplying by π/180. The sin function will still give the correct value in either mode, but the term ½r²θ will be wrong if θ is in degrees.
Mistake: Confusing a segment with a sector and forgetting to subtract the triangle.
Correction: A sector is the full pie-slice from the center. A segment excludes the triangular part, so you must subtract ½r² sin θ from the sector area.

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