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sq units
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units
The Sector Area Calculator computes the area of a circular sector — the 'pie slice' shaped region bounded by two radii and the arc between them. Given the radius and central angle in degrees, this calculator applies the formula $$A = \frac{1}{2}r^2\theta$$ (where $$\theta$$ is in radians) to determine the enclosed area.
A sector is one of the most commonly encountered geometric shapes in both mathematics and everyday life. Every slice of pizza, every wedge of pie, and every segment of a radar display is a sector. Sectors appear in pie charts, fan blades, windshield wiper sweep areas, and architectural designs featuring curved walls with radial boundaries.
The sector area formula has an intuitive derivation. A full circle has area $$\pi r^2$$ and corresponds to an angle of $$2\pi$$ radians (360°). A sector with central angle $$\theta$$ occupies the fraction $$\frac{\theta}{2\pi}$$ of the full circle. Therefore, sector area = $$\frac{\theta}{2\pi} \times \pi r^2 = \frac{1}{2}r^2\theta$$. This elegant formula shows that sector area depends linearly on the angle and quadratically on the radius.
In engineering and design, sector area calculations are routine. Mechanical engineers compute the sweep area of rotating components such as radar antennas, sprinklers, and windshield wipers. Civil engineers calculate land area for pie-shaped lots at cul-de-sacs and roundabout exits. Agricultural engineers determine the coverage area of rotating irrigation systems — center-pivot irrigators create circular fields, and each nozzle covers a sector.
In data visualization, pie charts divide a circle into sectors whose areas are proportional to the data values they represent. Understanding sector area is essential for creating accurate and readable pie charts. A category representing 25% of the data gets a 90° sector (one-quarter of 360°), and its area is exactly one-quarter of the full circle.
The calculator also returns the arc length of the sector, which is the curved boundary portion. This is useful when you need to know the length of the curved edge — for instance, the length of crust on a pizza slice, or the length of curved trim on an architectural element.
This tool handles the degree-to-radian conversion automatically. Enter the radius and angle in degrees, and receive both the sector area and arc length instantly. Whether you are a student studying circle geometry, a professional designing circular components, or simply curious about the area of a slice of your favorite circular food, this calculator provides quick and accurate results.
The Sector Area Calculator uses the standard formula for the area of a circular sector:
$$A = \frac{1}{2}r^2\theta$$
Where:
Degree to radian conversion:
$$\theta = \frac{\alpha \times \pi}{180}$$
Alternatively, using degrees directly:
$$A = \frac{\alpha}{360} \times \pi r^2$$
Both formulations are mathematically equivalent.
Arc length is also computed as:
$$s = r\theta$$
Steps:
The Sector Area is the total two-dimensional region enclosed by the two radii and the arc. It is measured in square units corresponding to the radius units. A 90° sector of a circle with radius 10 has area $$\frac{1}{4} \times \pi \times 100 \approx 78.54$$ square units — exactly one-quarter of the full circle's area.
The Arc Length gives the length of the curved boundary of the sector. Together with the two straight radii, this defines the full perimeter of the sector, which equals $$2r + s$$ (two radii plus the arc).
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θ = 90° = π/2 rad. Area = 0.5 × 100 × π/2 = 25π ≈ 78.54 sq units. Arc = 10 × π/2 ≈ 15.71 units. This is exactly one-quarter of the circle's total area (π × 100 ≈ 314.16).
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θ = 120° = 2π/3 rad. Area = 0.5 × 225 × 2π/3 = 75π ≈ 235.62 m². The sprinkler covers about 236 square meters. Arc length ≈ 31.42 m along the outer edge.
A sector is bounded by two radii and an arc (the 'pie slice' shape). A segment is bounded by a chord and an arc (the region between a chord and the arc). The sector includes the triangular region near the center; the segment does not.
If you know the arc length $$s$$ and radius $$r$$, then $$\theta = s/r$$ and $$A = \frac{1}{2}rs$$. This formula is particularly convenient because it avoids explicit angle calculations.
The perimeter of a sector is the sum of the two radii and the arc length: $$P = 2r + s = 2r + r\theta$$. For a 90° sector with radius 10, $$P = 20 + 15.71 = 35.71$$ units.
Yes. A sector with a central angle greater than 180° is called a major sector. The formula works identically. A 270° sector has three-quarters of the circle's area.
In a pie chart, each category's sector has a central angle proportional to its share of the total. If a category is 30% of the data, its sector angle is $$0.30 \times 360° = 108°$$, and its area is 30% of the circle's total area.
A 360° sector is the entire circle. The formula gives $$A = \frac{1}{2}r^2 \times 2\pi = \pi r^2$$, which is exactly the area of the full circle, confirming the formula's consistency.
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