Enter values to see results
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sq units
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sq units
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sq units
Enter values to see results
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sq units
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sq units
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sq units
The Segment Area Calculator computes the area of a circular segment — the region between a chord and the arc it subtends. This is a more advanced circle calculation than the basic sector, and it appears frequently in engineering, architecture, and applied mathematics.
A circular segment is formed when a straight line (chord) cuts across a circle, creating two regions. The smaller region (for chords that do not pass through the center) is called the minor segment, and the larger region is the major segment. This calculator handles both cases, depending on the central angle: angles less than 180° correspond to the minor segment, and angles greater than 180° correspond to the major segment.
The formula for segment area is derived by subtracting the triangular area from the sector area. The sector (pie slice) bounded by two radii and the arc has area $$A_{\text{sector}} = \frac{1}{2}r^2\theta$$. The isosceles triangle formed by the two radii and the chord has area $$A_{\text{triangle}} = \frac{1}{2}r^2\sin\theta$$. The segment is what remains:
$$A_{\text{segment}} = \frac{1}{2}r^2(\theta - \sin\theta)$$
This elegant formula captures the geometry perfectly. When $$\theta$$ is small, $$\sin\theta \approx \theta$$, so the segment area approaches zero — a very thin sliver. When $$\theta = \pi$$ (180°), $$\sin\theta = 0$$, so the segment area equals the sector area, which is a semicircle. This makes intuitive sense: a chord through the center (a diameter) creates a segment that is exactly a semicircle.
Segment area calculations are important in several practical contexts. In hydraulic engineering, the cross-sectional area of water flowing through a partially filled circular pipe is a circular segment. Knowing this area is essential for calculating flow rates using the Manning equation or other open-channel flow formulas. In structural engineering, circular segments appear in arch analysis and in the design of curved beams and vaults.
In optics, circular segments describe the illuminated region when a circular aperture is partially obscured. In astronomy, the phases of the Moon can be described geometrically using circular segments and lunes. In everyday construction, when a circular window or arch intersects a straight wall, the visible area is a segment.
The calculator also displays the sector area and triangle area separately, allowing you to see how the segment area is decomposed. This breakdown is useful for educational purposes and for verifying the calculation. The relationship $$A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}}$$ is clearly visible in the results.
Enter the radius and central angle in degrees, and the calculator handles the conversion to radians internally. All three areas are returned to four decimal places of precision.
The Segment Area Calculator decomposes the problem into two parts:
Step 1: Sector Area
$$A_{\text{sector}} = \frac{1}{2}r^2\theta$$
Step 2: Triangle Area
The isosceles triangle formed by the two radii and the chord has area:
$$A_{\text{triangle}} = \frac{1}{2}r^2\sin\theta$$
Step 3: Segment Area
$$A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} = \frac{1}{2}r^2(\theta - \sin\theta)$$
Where:
For angles greater than 180°, the formula automatically computes the major segment, as $$\sin\theta$$ becomes negative, making $$(\theta - \sin\theta) > \theta$$.
The Segment Area is the crescent-shaped region between the chord and the arc. For small angles, this is a thin sliver; for angles approaching 180°, it approaches a semicircle; for angles beyond 180°, it becomes the major segment (more than half the circle).
The Sector Area and Triangle Area are shown for reference. You can verify that Segment = Sector - Triangle. This decomposition helps build geometric intuition about the relationship between these three areas.
Inputs
Results
θ = π/2. Sector = 0.5 × 100 × π/2 = 78.54. Triangle = 0.5 × 100 × sin(π/2) = 0.5 × 100 × 1 = 50.00. Segment = 78.54 - 50.00 = 28.54 sq units.
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Results
θ = π/3. Sector = 0.5 × 64 × π/3 = 33.51. Triangle = 0.5 × 64 × sin(60°) = 32 × 0.866 = 27.71. Segment = 33.51 - 27.71 = 5.80 sq units — a relatively thin region.
A sector is bounded by two radii and an arc (pie-slice shape). A segment is bounded by a chord and an arc (the region between a straight cut and the curved boundary). The segment equals the sector minus the triangle formed by the two radii and the chord.
As the central angle approaches 0°, the segment area approaches zero because the arc nearly coincides with the chord. Mathematically, for very small $$\theta$$, $$\theta - \sin\theta \approx \frac{\theta^3}{6}$$, which vanishes as $$\theta \to 0$$.
A major segment is the larger region created when a chord divides a circle. It corresponds to a central angle greater than 180°. Its area equals the full circle area minus the minor segment area. This calculator handles major segments automatically when you enter an angle greater than 180°.
When a circular pipe is partially filled, the water cross-section forms a circular segment. The segment area determines the flow cross-section, which is needed to calculate volumetric flow rate using formulas like $$Q = Av$$ (flow rate = area × velocity).
Yes, if you know both the chord length $$c$$ and the radius $$r$$. The central angle can be found from $$\theta = 2\arcsin\left(\frac{c}{2r}\right)$$, then applied in the segment area formula. This requires $$c \leq 2r$$ (chord cannot exceed diameter).
At 180°, the chord is a diameter. $$\sin(\pi) = 0$$, so the segment area equals the sector area: $$\frac{1}{2}r^2\pi = \frac{\pi r^2}{2}$$, which is exactly a semicircle. The triangle collapses to zero area.
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