Find The Area Of The Sector Calculator

Easily calculate the area of a sector of a circle using our Area of a Sector Calculator. Enter the radius and the central angle below.

What is the Area of a Sector?

The area of a sector is the amount of space enclosed by two radii of a circle and the arc that connects them. Imagine a slice of pizza or a piece of pie; the shape of that slice is a sector of the circle from which it was cut. The Area of a Sector Calculator helps you find this area quickly.

A sector is defined by its radius (the radius of the circle) and its central angle (the angle between the two radii at the center of the circle). To find the area of the sector, you need to know these two values. Our Area of a Sector Calculator uses these inputs to give you the precise area.

Who should use the Area of a Sector Calculator?

• Students: Learning geometry and trigonometry concepts.
• Engineers and Architects: Designing circular elements or calculating material usage for curved sections.
• Designers: Working with circular layouts and patterns.
• Hobbyists: Engaging in crafts or projects involving circular shapes.

Common Misconceptions

One common misconception is confusing the area of a sector with the area of a segment. A segment is the region between an arc and its chord, while a sector is the region between an arc and two radii. Another point of confusion can be the units of the angle – always ensure you know whether the angle is in degrees or radians when using a formula or our Area of a Sector Calculator.

Area of a Sector Formula and Mathematical Explanation

The area of a sector is a fraction of the total area of the circle. The fraction is determined by the central angle of the sector compared to the total angle in a circle (360 degrees or 2π radians).

The area of a full circle is given by the formula: A_circle = πr²

If the central angle θ is given in degrees, the area of the sector is:

A_sector = (θ / 360) * πr²

If the central angle θ is given in radians, the area of the sector is:

A_sector = (θ / 2π) * πr² = (1/2) * θr² = (θr²)/2

Our Area of a Sector Calculator uses the appropriate formula based on the angle unit you select.

Variables Table

Variable	Meaning	Unit	Typical Range
Asector	Area of the Sector	Square units (e.g., m², cm²)	≥ 0
r	Radius of the circle	Length units (e.g., m, cm)	> 0
θ	Central angle	Degrees or Radians	0 to 360 (degrees), 0 to 2π (radians)
π	Pi (approx. 3.14159)	Dimensionless	3.14159…

Variables used in the Area of a Sector calculation.

Practical Examples (Real-World Use Cases)

Example 1: Pizza Slice

Imagine a circular pizza with a radius of 18 cm. It is cut into 8 equal slices. What is the area of one slice?

The total angle is 360 degrees, so each slice has a central angle of 360 / 8 = 45 degrees. The radius r = 18 cm.

Using the Area of a Sector Calculator or the formula: Area = (45 / 360) * π * (18)² = (1/8) * π * 324 ≈ 127.23 cm².

Each slice has an area of approximately 127.23 square centimeters.

Example 2: Garden Sector

A circular garden has a radius of 10 meters. A sector of this garden with a central angle of 1.2 radians is to be paved.

Here, r = 10 m and θ = 1.2 radians.

Using the Area of a Sector Calculator or the formula for radians: Area = (1.2 * 10²) / 2 = (1.2 * 100) / 2 = 120 / 2 = 60 m².

The area to be paved is 60 square meters.

How to Use This Area of a Sector Calculator

1. Enter the Radius (r): Input the radius of the circle from which the sector is taken. This value must be positive.
2. Enter the Central Angle (θ): Input the angle formed at the center by the two radii. Ensure it's non-negative.
3. Select Angle Unit: Choose whether the angle you entered is in degrees or radians from the dropdown menu.
4. View Results: The calculator will instantly display the Area of the Sector, the angle in radians (if you input degrees), the arc length, and the total area of the circle. The formula used will also be shown.
5. Analyze Chart: The chart visually represents the sector within the circle based on your inputs.
6. Reset: Click the "Reset" button to clear the inputs and results to their default values.
7. Copy Results: Click "Copy Results" to copy the main result, intermediate values, and input parameters to your clipboard.

The Area of a Sector Calculator provides quick and accurate results, helping you understand the relationship between radius, angle, and sector area.

Key Factors That Affect Area of a Sector Results

• Radius (r): The area of the sector is proportional to the square of the radius (r²). Doubling the radius increases the area fourfold, keeping the angle constant.
• Central Angle (θ): The area of the sector is directly proportional to the central angle. Doubling the angle doubles the area, keeping the radius constant.
• Angle Unit: The formula used depends on whether the angle is in degrees or radians. The calculator handles this conversion if needed. Using the wrong unit in a manual calculation will give a very different result.
• Value of Pi (π): The accuracy of the result depends on the precision of Pi used. Our Area of a Sector Calculator uses a high-precision value of Math.PI.
• Measurement Accuracy: The accuracy of the calculated area is dependent on the accuracy of the input radius and angle measurements.
• Full Circle vs. Sector: The sector area is always a fraction (θ/360 or θ/2π) of the total circle's area (πr²).

Frequently Asked Questions (FAQ)

What is a sector of a circle?

A sector is a part of a circle enclosed by two radii and the arc between them, resembling a slice of pie.

How is the area of a sector different from the area of a segment?

A sector is bounded by two radii and an arc, while a segment is bounded by a chord and an arc. The Area of a Sector Calculator finds the area of the pie slice shape.

What units are used for the area of a sector?

The area is measured in square units, such as square meters (m²), square centimeters (cm²), etc., depending on the unit of the radius.

Can the central angle be greater than 360 degrees or 2π radians?

While geometrically a simple sector is usually within 0-360 degrees, you can input larger angles; the calculator will compute the area corresponding to that angle (e.g., representing multiple wraps).

What if the angle is 0?

If the angle is 0, the area of the sector is 0.

What if the angle is 360 degrees or 2π radians?

If the angle is 360 degrees (or 2π radians), the area of the sector is equal to the area of the full circle.

Does this calculator find the arc length?

Yes, the Area of a Sector Calculator also displays the arc length as an intermediate result.

Can I use negative values for radius or angle?

The radius must be positive, and the angle is typically considered non-negative for area calculations. The calculator will show an error for negative radius and will treat negative angles based on their positive magnitude for area, although the chart might look different.

Related Tools and Internal Resources

• Arc Length Calculator: Calculate the length of the arc of a sector.
• Circle Calculator: Find the area, circumference, and diameter of a circle.
• Radians to Degrees Converter: Convert angles from radians to degrees.
• Degrees to Radians Converter: Convert angles from degrees to radians.
• Area of a Segment Calculator: Calculate the area of a circle segment.
• Geometry Calculators: Explore more calculators related to geometric shapes.