Partial Area of a Circle Calculator

A partial area of a circle calculator computes the area of a sector (a pie-slice shape) or a segment (the region between a chord and the arc) when you enter the circle’s radius and the central angle. You get the area in square units, along with the arc length and chord length if needed. This tool is useful for geometry homework, engineering layouts, and any project involving circular sections.

Formula

For a sector (the slice from the center to the arc):

Area = (θ / 360) × π × r²

Where:

• θ (theta) = central angle in degrees
• r = radius (same unit as the area’s linear dimension)
• π ≈ 3.14159

For a segment (the area between the chord and the arc, not including the triangle):

Area = (θ / 360) × π × r² − ½ × r² × sin(θ)

Here sin(θ) uses the angle in radians. If your calculator is in degree mode, convert: sin(θ°) = sin(θ × π / 180).

Arc length (for either shape) = (θ / 360) × 2πr

Chord length = 2 × r × sin(θ / 2) (again, θ in radians for the sine)

Worked Example

Using the default values from the calculator: radius = 5 m, central angle = 60°, and shape = sector.

1. Convert the angle to a fraction of the full circle: 60 / 360 = 1/6.
2. Compute the full circle area: π × 5² = 25π ≈ 78.54 m².
3. Multiply by the fraction: (1/6) × 78.54 ≈ 13.09 m².

The calculator shows 13.09 m². For the same inputs as a segment, the area would be:

13.09 − ½ × 25 × sin(60°) = 13.09 − 12.5 × 0.8660 ≈ 13.09 − 10.83 = 2.26 m².

When to Use a Partial Circle Area

You need a partial area whenever the shape is not a full circle. Common situations:

• Cutting a circular pizza or cake into equal slices – each slice is a sector.
• Designing an arched window or doorway – the glass or frame may be a segment.
• Calculating the area of a circular garden bed that is partly blocked by a path.
• Working with gears, pulleys, or circular machine parts that have a missing section.
• Estimating the area of a round table covered by a cloth that hangs in a segment.

In geometry problems, “partial area” usually means either a sector or a segment. The calculator lets you pick which one you need, because they have different formulas.

How to Read the Results

The main output is the area in square units (m², cm², ft², etc.). The arc length is the curved edge of the sector or segment; the chord is the straight line connecting the two endpoints of the arc. For a sector, the straight edges are the two radii, so the chord is not part of the boundary. For a segment, the chord is the flat base.

If you are measuring a real object, make sure the radius and the angle are measured from the same center point. The angle must be the central angle – the angle at the center of the circle, not an angle at the edge.

Common Mistakes and How to Avoid Them

• Mixing degrees and radians: The calculator expects degrees. If you have radians, multiply by 180/π to convert. For example, π/3 radians = 60°.
• Using the diameter instead of the radius: The radius is half the diameter. If you measure across the whole circle, divide by 2 first.
• Forgetting to subtract the triangle for a segment: The segment area is the sector minus the isosceles triangle formed by the two radii and the chord. If you only need the sector, choose “sector”.
• Rounding too early: Do the full calculation, then round to the desired number of decimals. Rounding intermediate steps can change the final answer by a few percent.

Limitations of the Formula

The formulas assume a perfect circle and a straight chord. Real-world objects may have slight irregularities, but for most practical purposes the error is negligible. The angle must be less than 360°; if you enter 360°, you get the full circle area. For a segment, the angle must be between 0° and 180° for the chord to be the shorter arc; if the angle is greater than 180°, the segment is the larger part of the circle (major segment), and the formula still works but the triangle subtraction becomes an addition (the area is the sector plus the triangle). The calculator here uses the standard formula that works for any angle, but the result for angles > 180° may be counterintuitive – the area is larger than the sector alone.

Alternatives and Related Calculations

If you need the area of a full circle, use A = πr². For a semicircle (half circle), use A = ½πr². For a quarter circle, use A = ¼πr². The partial area formula generalizes these: any fraction of a circle is (θ/360) × πr².

You might also need the perimeter of a sector (arc length + 2r) or the perimeter of a segment (arc length + chord). Those are easy to compute once you have the arc length and chord length from the calculator.

Frequently Asked Questions

What is the difference between a sector and a segment?

A sector is the region bounded by two radii and the arc between them – it looks like a pizza slice. A segment is the region bounded by a chord and the arc – it looks like a crescent or a piece of an orange slice without the point. The segment is always smaller than the sector for the same angle because it excludes the triangle formed by the two radii.

Can I use this calculator for angles in radians?

No, the calculator expects degrees. If you have radians, convert to degrees by multiplying by 180/π. For example, 1 radian ≈ 57.2958°. You can also convert the angle to degrees manually before entering it.

What if my angle is greater than 180 degrees?

For a sector, the formula works for any angle up to 360°. For a segment, if the angle is between 180° and 360°, you are dealing with a major segment (the larger part of the circle). The formula still gives the correct area, but you might want to think of it as the full circle minus the minor segment. The calculator will produce a positive number, but the interpretation changes.

How do I find the radius if I only know the chord length and the angle?

You can rearrange the chord formula: r = chord / (2 × sin(θ/2)). Make sure the angle is in radians for the sine. For example, if the chord is 5 m and the angle is 60°, then r = 5 / (2 × sin(30°)) = 5 / (2 × 0.5) = 5 m. Then use that radius in the area formula.

Why does the segment area become negative for large angles?

It shouldn’t with the correct formula. For angles greater than 180°, the term ½r²sin(θ) becomes negative (since sine is negative between 180° and 360°), so subtracting it actually adds to the sector area. The result is always positive. If you see a negative number, you likely used the wrong sign or the angle in degrees instead of radians in the sine.

Can I use this for a circle with a hole?

No, that would be an annulus (ring) area, which is the difference between two full circles. For a partial ring (like a section of a washer), you would subtract the inner sector from the outer sector. That is a more advanced calculation not covered here.

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in r | Radius | m | 5 | 0
in a | Central angle | degrees | 60 | 0
sel type | Shape | Sector=1, Segment=2
out area | Area | m² | (type==1 ? (a/360)*pi*r^2 : (a/360)*pi*r^2 - 0.5*r^2*sin(a*pi/180)) | 2
out arclen | Arc length | m | (a/360)*2*pi*r | 2
out chord | Chord length | m | 2*r*sin(a*pi/360) | 2