Area of a Circular Sector Calculator

This calculator works out the area of a circular sector from the radius and either the central angle (in degrees or radians) or the arc length. You enter the radius and one other value, and you get the sector area in square units, plus the arc length and chord length for reference.

Formula

The area of a circular sector is the fraction of the full circle’s area that the sector covers. If the angle is given in degrees, the formula is:

A = (θ / 360) × π × r²

where:

• A = sector area (same unit squared as the radius)
• θ = central angle in degrees
• r = radius (any length unit)
• Ï€ ≈ 3.14159

If the angle is in radians, use the simpler form:

A = (1/2) × r² × θ

with θ in radians. Both formulas give the same result; the calculator above converts degrees to radians automatically when needed.

Worked Example

Using the default values: radius = 5 m, angle = 60°.

1. Convert the angle to radians: 60° × (π/180) = π/3 ≈ 1.0472 radians.
2. Apply the radian formula: A = 0.5 × 5² × 1.0472 = 0.5 × 25 × 1.0472 = 13.09 m².
3. Check with the degree formula: (60/360) × π × 25 = (1/6) × 78.5398 = 13.09 m².

The calculator shows 13.09 m² as the headline answer, matching the step-by-step result.

How to Use the Calculator

Enter the radius in any unit (meters, feet, inches, etc.) and the central angle in degrees or radians. The calculator returns:

• Sector area – the main result, in square units of the radius (e.g., m² if radius is meters).
• Arc length – the curved distance along the sector’s edge.
• Chord length – the straight line between the two endpoints of the arc.

For a full circle (360°), the sector area equals πr². For a semicircle (180°), it is half that. The angle must be between 0 and 360 degrees (or 0 and 2π radians); the calculator rejects values outside that range.

When to Use a Sector Area

You need sector area in practical geometry and design tasks:

• Calculating the area of a slice of pie or pizza (if you know the radius and angle).
• Finding the amount of material for a circular fan-shaped piece of fabric, metal, or wood.
• Determining the area of a portion of a circular lawn or garden bed.
• Working with gear teeth, cam profiles, or any part shaped like a pie slice.

In landscaping, for example, a sprinkler that covers a 90° arc with a 10 m throw waters a sector of area (90/360)×π×10² ≈ 78.54 m². That is a quarter of the full circle’s area.

Common Mistakes and How to Avoid Them

The most frequent error is mixing units or using degrees in the radian formula. Always check whether your angle is in degrees or radians before applying a formula. Another mistake is using the diameter instead of the radius – the formula requires the radius, which is half the diameter.

Rounding too early can also skew results. For a 60° sector with r=5, rounding π to 3.14 gives 13.08 m² instead of 13.09 m² – a small difference, but for large radii it grows. Keep at least 4-5 significant figures during intermediate steps.

Finally, remember the angle must be less than 360° for a single sector. If you need the area of a shape that is a full circle minus a sector, subtract the sector area from the full circle area.

Limitations of the Formula

The sector area formula assumes a perfect circle and a straight-line radius. It does not apply to ellipses, irregular curves, or sectors where the boundary is not a circular arc. For such shapes, numerical integration or CAD software is needed.

Also, the formula gives the area of the flat, two-dimensional sector. If you are working with a three-dimensional object like a cone (where the sector is the lateral surface), you need additional formulas for surface area or volume.

Frequently Asked Questions

Can I use this calculator with the angle in radians?

Yes. Select “Radians� from the angle mode dropdown, then enter the angle in radians (e.g., 1.0472 for 60°). The calculator converts internally and returns the correct area.

What if I only know the arc length, not the angle?

You can still find the area. The arc length L = rθ (with θ in radians). So θ = L/r. Then the area = (1/2)r²θ = (1/2)rL. This calculator does not take arc length as input directly, but you can compute θ = L/r and enter that.

Does the unit of the radius affect the area?

Yes, the area is in square units of the radius. If you enter the radius in feet, the area is in square feet. If you enter meters, the area is in square meters. The calculator does not convert units – you must use consistent units.

Why is the angle limited to 360 degrees?

A sector is defined as part of a circle, so its central angle cannot exceed a full revolution (360° or 2π radians). If you need the area of a shape that wraps around more than once, you would add multiple sectors.

How accurate is the result?

The calculator uses a high-precision value of π (to about 15 digits) and rounds the final answer to two decimal places. For most practical purposes, this is more than sufficient. If you need more precision, you can increase the decimal places in the output settings, but two decimal places is typical for construction and design work.

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in r | Radius | m | 5 | 0 | 100000
sel anglemode | Angle in | Degrees=1, Radians=2 | 1
in theta | Central angle | ° | 60 | 0 | 360
out area | Sector area | m² | 0.5 * r^2 * (theta * (pi/180)) | 2
out arclen | Arc length | m | r * (theta * (pi/180)) | 2
out chord | Chord length | m | 2 * r * sin((theta * (pi/180))/2) | 2