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Sector Area Calculator

Calculate the area of a circular sector given the radius and central angle.

By Christopher FloiedPublished Updated

Minimum: 0

The angle of the sector at the center of the circle

Results

Sector Area

52.3599 sq units

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Sector Area Calculator. Follow the units and input format printed next to each field. Where a unit selector is available, choose the units that match your data.

2

Review your inputs

Double-check that all values are correct and match the units or format shown for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Sector Area Calculator computes the output in your browser and displays the results, with units clearly labeled, as soon as they are ready — no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Sector Area Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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Arc Length Calculator

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Annulus Area Calculator

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Chord Length Calculator

Calculate the length of a chord in a circle given the radius and the central angle subtended by the chord. Uses the formula c = 2r*sin(theta/2), essential for geometry, engineering, surveying, and arc-based constructions in architecture and design.

Area of a Circle Calculator

Calculate the area of a circle given its radius. Uses the formula A = πr² for precise results.

About Sector Area Calculator

A sector is the region of a circle enclosed between two radii and the arc connecting them — like a slice of pie. The area of a sector is proportional to the central angle: a larger angle means a larger slice. The formula A = ½r²θ (θ in radians) follows from the proportion of the full circle that the sector represents. Sector area calculations are used in pie charts, fan blade design, sprinkler coverage, camera field-of-view analysis, and many other applications. This is also the geometric interpretation behind the formula for radian measure and connects to the integration formula for areas in polar coordinates. This calculator converts degrees to radians automatically and computes the sector area for any combination of radius and angle.

The Math Behind It

The sector area formula A = ½r²θ can be derived as a fraction of the total circle area: A = (θ/2π) × πr² = ½r²θ. In degrees, this becomes A = (πr²θ°)/360. This formula is the 2D analog of the arc length formula s = rθ. The relationship between sector area and arc length is A = ½rs, where s is the arc length. In polar coordinates, the area enclosed by a curve r(θ) from angle α to β is A = ½∫ₐᵝ r(θ)² dθ, which is a direct generalization of the sector area formula. For a sector with area A and perimeter P = 2r + rθ, the maximum area for a fixed perimeter occurs when θ = 2 radians (approximately 114.6°). The segment area (the region between a chord and the arc) equals the sector area minus the triangle area: A_segment = ½r²(θ − sin θ).

Formula Reference

Sector Area

A = ½ r² θ

Variables: r = radius, θ = central angle in radians

Worked Examples

Example 1: Pizza slice

A pizza with radius 7 inches is cut into 8 equal slices. Find the area of one slice.

Step 1:θ = 360° / 8 = 45° = π/4 radians
Step 2:A = ½ × 7² × (π/4)
Step 3:A = ½ × 49 × 0.7854

A ≈ 19.24 square inches

Common Mistakes & Tips

  • !Forgetting to convert the angle to radians before applying A = ½r²θ.
  • !Confusing sector area with segment area (segment = sector minus triangle).
  • !Using the diameter instead of the radius.

Related Concepts

Arc Length

The length of the curved edge of the sector.

Area of a Circle

The area of a full sector (360°): A = πr².

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Arc Length Calculator

Frequently Asked Questions

What is the difference between a sector and a segment?

A sector is bounded by two radii and an arc (like a pie slice). A segment is bounded by a chord and the arc it cuts off. Segment area = sector area − triangle area.

How do I find the angle if I know the area?

Rearrange: θ = 2A/r² (radians) or θ° = 360A/(πr²) (degrees).

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