Sector Area Calculator

Radius (r)

Central angle (θ)

Sector area (A)

Also show the segment and chord

Split the slice into the triangle and the segment cut off by the chord.

Also show the major (leftover) sector

The larger part of the circle left once your slice is removed.

Enter any two of the radius, the central angle, and the sector area, and the calculator fills in the third for you, along with the arc length, perimeter, and more.

Please input values into the fields

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Show the slice-vs-circle pie

How much of the whole circle your sector covers.

Show how the area scales with the radius

A bar chart of how the sector area grows as the radius grows.

A sector is a part of a circle bounded by two radii and the arc that joins them. A typical example would be a slice of pizza or pie, viewed from above.

This sector area calculator lets you find the area of a sector from any combination of circle radius and central angle measurements. Enter any two values and this tool will calculate the remaining ones including arc length, perimeter, chord, and sector as a percentage of the whole circle.

What is a sector of a circle?

A sector is an area bounded by two radii and the arc between them. The central angle is the angle formed by these two radii at the center of the circle.

The smaller part is called the minor sector while the larger part is called the major sector. When both the sectors are added together they make up the whole circle.

A sector is different from a segment. A segment is the small area cut off by a straight line (chord). The area of the triangle formed by the two radii is subtracted from the area of the sector to get the segment. By opening up the segment plane in the tool, both can be shown at once.

Formula for area of a sector

If the central angle is measured in degrees, then the area of the sector is the product of the ratio of the measure of the arc to 360 and the area of the circle.

A = \dfrac{\theta}{360^\circ} \times \pi r^2

In this formula, A is the area of the sector, r is the radius and θ is the central angle. Dividing θ by 360 gives the proportion of the whole circle that the sector covers.

If the angle is measured in radians, then the formula becomes even simpler since a full circle is 2π radians.

A = \tfrac{1}{2} r^2 \theta

The results obtained with these two formulas are identical. This calculator automatically converts between degrees and radians, so the angle can be entered in either unit.

Method for calculating the area of a sector.

You set the radius and central angle. You square the radius, multiply it by pi, then by the value divided by the degrees of the angle (360).

For example, consider a circle with a radius of 7 centimeters and a central angle of 45 degrees. Since 45 degrees is one-eighth of the entire circle, this sector will have one-eighth of the total area of the circle.

A = \dfrac{45}{360} \times \pi \times 7^2 = \dfrac{1}{8}\times 49\pi \approx 19.24 \ \text{cm}^2

If you know the area and the angle, but want to find the radius instead, rearrange the formula to solve for r, then take the square root.

r = \sqrt{\dfrac{360\,A}{\pi\,\theta}}

If you know the area and radius and want to determine the central angle, re-formulate the formula.

\theta = \dfrac{360\,A}{\pi r^2}

You do not have to decide which formula to use yourself. If you enter any two of the three values, the calculator automatically transforms the corresponding formula.

Arc length, perimeter, chord.

The two inputs can be used to find other data about the sector. The arc length is the length of the curved side and is given in radians, which is found by multiplying the radius by the angle.

L = r\theta \qquad P = 2r + L

The perimeter of a sector is made up of the curved arc length and two straight radii. A chord is a straight line that connects the two endpoints of an arc, and its length is calculated as the product of twice the radius and the sine of half the angle.

c = 2r\sin\!\left(\tfrac{\theta}{2}\right)

Examples of calculations:

The following table shows the area of sectors for common combinations of radius and angle, with results rounded to two decimal places. Note that the area depends on the square of the radius, so it increases rapidly as the radius gets larger. In contrast, the area changes proportionally to the angle.

Radius	Angle	Sector area
7 cm	45 degrees	19.24 sq cm
10 cm	90 degrees	78.54 sq cm
6 ft	120 degrees	37.70 sq ft
50 m	72 degrees	1570.80 sq m

A sector with a central angle of 90 degrees is one quarter of the circle, so it takes up exactly one fourth of the area. A sector with a central angle of 180 degrees is half of the circle and takes up exactly half of the area. A sector with a central angle of 60 degrees takes up one sixth of the area while a sector with a central angle of 72 degrees takes up one fifth of the area, which corresponds to the size of each wedge in a pie chart divided into five equal wedges.

Special cases (half circle and quarter circle)

Certain angles correspond to special shapes, each with its own name. A quarter circle is also called a quadrant and has an angle of 90 degrees. Its area corresponds exactly to one fourth of the total area of the circle.

A semicircle has an angle of 180 degrees and its area is exactly half the total area of a circle. The calculator will show these special cases when they apply.

Applications of area calculation for sectors

In objects or situations where an area extends from the center outward in a sector shape, calculating the area of sectors can be useful. A lawn sprinkler that sprays water at a 120-degree angle covers a sector-shaped region, and its watering area is equal to the calculated area of the sector based on the radius distance.

Designers determine the size of each sector in a pie chart, windshield wipers clean a sector-shaped pane of glass, and surveyors use radius and angle to represent a parcel of land. Even cooks sometimes compare slices of pizza this way.

Tips for accurate results

Measure the radius accurately from the exact center of the circle and make sure that the angle is stated in your desired unit, whether degrees or radians. The units for length and area must match. If you measure the radius in centimeters, for example, then the area will be calculated in square centimeters.

Because area depends on the square of the radius, a small error in the radius leads to a large error in the area. So measure your radius as accurately as possible.

Frequently asked questions

What is the formula for calculating the area of a sector?

If the area is expressed in degrees, it can be calculated using the formula A = (theta/360)*pi*r^2 where θ is the central angle and r is the radius. When the area is expressed in radians, the formula simplifies to A = 0.5*r squared*theta. With a radius of 7cm and an angle of 45 degrees, the result is approximately 19.24 square centimeters.

How do you find the central angle of a sector?

The formula for area is rearranged to be: θ = 360 multiplied by the area, divided by pi * r squared. In this calculator you can simply enter in the radius and the area and leave the angle field blank to calculate the angle.

Can you determine the radius if you know the area and angle?

Yes. The radius is the square root of the product of area and 360, divided by the product of pi and angle. If you enter the area and the angle and leave the field for the radius empty, the tool will calculate its value.

What is the difference between a sector and a segment of a circle?

A sector is a complete wedge-shaped area bounded by two radii and an arc, similar to a slice of pizza. A segment is a smaller area bounded by a straight line (chord) between the ends of the arc. If you subtract from the sector the triangle formed by the two radii, you get the segment. You can see both at once if you open the "segment" layer.

How do you convert an angle to radians so that it can be used in the sector formula?

Multiply the degrees by pi and divide it by 180. So, 180 degrees is equal to π radians, and 90 degrees is equal to π/2 radians. You don't have to do this conversion manually; just change the units of your angle measure to radians and enter that value.

What fraction of a circle is a sector with an angle measure of 60 degrees?

Out of the 360 degrees, sixty degrees is one sixth of the entire circle which is about 16.7%. For any angle inputted, this calculator will show you what percentage it makes up of the whole circle.

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Circle Area CalculatorEnter any one of a circle's radius, diameter, circumference, or area and get the other three instantly, plus an optional sector area and arc length.

Circle Perimeter CalculatorEnter any one of a circle's radius, diameter, perimeter, or area and get the other three instantly, plus an optional sector arc length and sector perimeter.

Area CalculatorFind the area of a square, rectangle, triangle, circle, sector, ellipse, trapezoid, parallelogram, or rhombus. Pick a shape, enter its measurements, and get the area and perimeter in any unit.

Arc Length CalculatorFind arc length from any two of the central angle, radius, arc length, or sector area, plus the chord, diameter, and major arc.

Square Area CalculatorEnter any one of a square's side, diagonal, perimeter, or area and get the other three instantly, plus optional border and tiling-cost layers.

Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Circular sector
Definition of a sector, the area and arc length formulas in degrees and radians.
Wikipedia: Circular segment
How a segment relates to a sector and its chord.
Math is Fun: Area of a Sector
Plain-language explanation of sectors, segments, and arcs.