Sector of a Circle Calculator

Calculate the area, arc length, and perimeter of a circle's sector. Quickly solve for missing variables like radius and central angle using degrees or radians.

Input Data

Quick Presets

What do you want to calculate?

Sector Area

Central Angle

Radius

Radius (r)

units

Central Angle (θ)

Decimal Precision

Rounding Decimal Places

Calculation Results

Enter the missing values to instantly solve for the sector properties.

Instantly Solve Circular Sector Properties

A sector is a portion of a circle enclosed by two radii and an arc—much like a slice of pie. Whether you are working on a geometry assignment, drafting an architectural layout, or calculating material usage, this sector of a circle calculator helps you instantly solve for missing dimensions. You can seamlessly calculate the sector area, arc length, and perimeter using either degrees or radians. If you already know the area, you can also use our reverse-calculation modes to easily find the missing radius or central angle.

Calculator Features

Multi-Mode Solving

Don't just calculate area. Easily switch modes to find the central angle or the exact radius if you already have the sector area.

Degrees & Radians

Instantly toggle between degree and radian angle measurements without needing to perform manual unit conversions.

Mathematical Breakdown

Stop guessing how the result was found. The tool provides a clear, step-by-step formula breakdown with your exact numbers substituted in.

Smart Angle Presets

Speed up your workflow using one-click presets for common geometric shapes like quarter circles (90°) and semicircles (180°).

How to Use the Sector Calculator

Select Your Calculation Mode

Use the toggle buttons at the top of the tool to choose what you want to find: the sector area, the central angle, or the radius.

Enter Known Measurements

Input your known variables into the fields provided. If you are entering an angle, ensure you select the correct unit (Degrees or Radians) from the dropdown menu.

Review Instant Results

The tool processes your data in real time, displaying the primary result alongside secondary metrics like arc length, perimeter, and the percentage of the full circle.

Check the Formula Breakdown

Scroll down to the 'Formula Used' section to see the exact mathematical equation applied to your specific calculation.

Sector Area Formula (Degrees)

                                A = (θ / 360) × πr²
                            

When your central angle is measured in degrees, the area (A) is calculated by finding the fraction of the circle the sector represents (θ / 360) and multiplying it by the total area of the circle (πr²). In this formula, r represents the radius and θ represents the central angle in degrees.

Sector Area Formula (Radians)

                                A = ½r²θ
                            

When your central angle is measured in radians, the formula simplifies significantly. The area (A) is equal to one-half multiplied by the radius squared (r²), multiplied by the central angle (θ). You must ensure your angle is strictly in radians to use this version of the formula.

Arc Length Formula

                                L = rθ (for radians) or L = (θ / 360) × 2πr (for degrees)
                            

The arc length (L) is the distance along the curved outer edge of the sector. Just like the area, the formula changes depending on your angle unit. If you are using radians, you simply multiply the radius by the angle. For degrees, you find the fraction of the circle (θ / 360) and multiply it by the full circumference (2πr).

📌 Watch Your Units and Terminology

A common mistake in geometry is using the degree formula while inputting a radian angle (or vice versa), which will result in a completely incorrect answer. Additionally, always remember that your final sector area is expressed in squared units (e.g., cm², in²), whereas the radius, arc length, and perimeter are expressed in linear units (e.g., cm, in).

Sector vs. Segment of a Circle

Feature	Option 1	Option 2
Visual Shape	Looks like a slice of pie or pizza.	Looks like a rounded cap or a bowl.
Boundaries	Enclosed by two radii and a curved arc.	Enclosed by a single straight chord and a curved arc.
Center Point Connection	Always connects perfectly to the exact center of the circle.	Does not connect to the center of the circle.
Calculation Complexity	Easily calculated as a fraction of the total circle's area.	More complex; requires subtracting a triangle's area from a sector's area.

Frequently Asked Questions

How do I find the central angle if I only know the sector area and radius?

To find the central angle in degrees, use the formula θ = (360 × Area) / (πr²). If you need the angle in radians, use θ = (2 × Area) / r². You can also simply select 'Central Angle' from the top toggle of our calculator, and it will reverse the math for you instantly.

What is the difference between the arc length and the sector area?

Arc length measures the physical distance along the curved outer boundary of the sector (a 1D linear measurement). Sector area measures the total 2D flat space enclosed inside the two radii and the arc (measured in square units).

How do I use this as an area of the sector of a circle calculator if I only know the diameter?

The diameter of a circle is exactly twice the length of the radius. Simply divide your given diameter by two to find the radius. Once you have the radius, enter it into the calculator along with your central angle to find the area.

Can a sector have an angle greater than 180 degrees?

Yes. A sector with a central angle less than 180 degrees is called a 'minor sector'. A sector with an angle greater than 180 degrees (but less than 360) is called a 'major sector'. The mathematical formulas for area and arc length apply equally to both.

Why does my manual math differ slightly from the calculator's result?

This discrepancy usually happens due to early rounding. If you round the value of Pi (π) to exactly 3.14, or if you round your intermediate steps before the final equation, your answer will lose accuracy. This tool uses high-precision floating-point math to ensure exact geometric results.

Guides & Resources for this Tool

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