Sector Area Calculator

How Sector Area Works

A sector is a region of a circle bounded by two radii and the arc between them, resembling a slice of pie. The area of a sector represents the fractional portion of a full circle's area corresponding to the central angle. According to Euclid's Elements (Book III), the area of a circular region is proportional to the central angle that subtends it. This principle has been applied in mathematics, engineering, and cartography for over 2,000 years. Modern applications include calculating land areas in surveying, designing fan blades and turbine components, creating pie charts and data visualizations, and computing the coverage area of sprinkler systems and radar sweeps. The sector area formula is one of the most frequently used geometric calculations in both academic and professional settings.

A full circle has a central angle of 360 degrees (or 2 pi radians), so a sector with a 90-degree angle covers exactly one-quarter of the circle's area. A 180-degree sector is a semicircle, and a 60-degree sector is one-sixth of the full circle. Understanding this proportional relationship makes sector calculations intuitive -- you are simply finding what fraction of the total circle your angle represents, then multiplying by the total area. The arc length calculator computes the curved edge of the sector, which is a closely related calculation.

The Sector Area Formula

The sector area can be calculated using either degrees or radians. In degrees: A = (theta / 360) x pi x r squared. In radians: A = (1/2) x r squared x theta. The variables are: A = sector area (in square units), r = radius of the circle, and theta = central angle (in degrees or radians). The arc length is: L = r x theta (in radians) or L = (theta / 360) x 2 x pi x r (in degrees). The sector perimeter (total boundary length) is: P = 2r + L (two radii plus the arc).

Worked example: Find the area of a sector with radius 10 cm and central angle 60 degrees. Using the degree formula: A = (60 / 360) x pi x 10 squared = (1/6) x pi x 100 = 52.36 square cm. The arc length is (60/360) x 2 x pi x 10 = 10.47 cm. The perimeter is 2(10) + 10.47 = 30.47 cm. To convert to radians: 60 degrees = pi/3 radians, so A = (1/2) x 100 x (pi/3) = 52.36 square cm -- the same result.

Key Terms

Sector: A region of a circle enclosed by two radii and the arc between their endpoints on the circle. It resembles a pie slice or pizza slice.

Segment: A region of a circle bounded by a chord and the arc it subtends. Unlike a sector, a segment does not include the center of the circle. The area of a segment equals the sector area minus the triangle area formed by the two radii and the chord.

Central Angle: The angle formed at the center of the circle by the two radii defining the sector. Measured in degrees (0-360) or radians (0 to 2 pi).

Arc Length: The distance along the curved edge of the sector. Calculated as r x theta (radians) or (theta/360) x 2 pi r (degrees).

Radian: The standard unit of angular measure in mathematics, defined as the angle subtended by an arc equal in length to the radius. One full revolution equals 2 pi radians (approximately 6.2832). To convert degrees to radians: radians = degrees x pi / 180.

Sector Properties Reference Table

The following table shows sector area and arc length for common angles with a radius of 10 units, demonstrating the proportional relationship between angle and area.

Practical Examples

Example 1 -- Sprinkler coverage: A lawn sprinkler rotates through a 120-degree arc with a reach of 15 feet. The watered area is: A = (120/360) x pi x 15 squared = (1/3) x pi x 225 = 235.6 square feet. This helps determine how many sprinklers are needed to cover a lawn.

Example 2 -- Pizza slice area: A 14-inch diameter pizza (radius 7 inches) cut into 8 equal slices. Each slice has a central angle of 360/8 = 45 degrees. Area per slice = (45/360) x pi x 7 squared = (1/8) x 153.94 = 19.24 square inches. Compare this with a 16-inch pizza cut into 10 slices: (36/360) x pi x 64 = 20.11 square inches per slice -- slightly larger despite more slices.

Example 3 -- Reverse calculation (finding angle from area): A sector has a radius of 8 cm and an area of 50 square cm. What is the central angle? Rearranging the formula: theta = (A x 360) / (pi x r squared) = (50 x 360) / (pi x 64) = 18,000 / 201.06 = 89.5 degrees. The regular polygon calculator uses similar angle-based area formulas for multi-sided shapes.

Tips for Sector Calculations

• Always check your angle unit. The most common error is mixing degrees and radians. If using the radian formula A = (1/2)r squared theta, the angle must be in radians. A 90-degree angle in the radian formula without conversion gives a wildly incorrect result.
• Convert degrees to radians by multiplying by pi/180. Convert radians to degrees by multiplying by 180/pi. Many calculators and programming languages default to radians.
• For the area of a segment (the region between a chord and an arc), subtract the triangle area from the sector area: Segment area = Sector area - (1/2) x r squared x sin(theta).
• To find the radius from area and angle: r = sqrt(2A / theta) when theta is in radians, or r = sqrt(360A / (theta x pi)) when theta is in degrees.
• Annular sectors (the region between two concentric circles and two radii) have area: A = (theta / 360) x pi x (R squared - r squared), where R is the outer radius and r is the inner radius. Use this for calculating the area of curved walkways, ring-shaped gardens, or elliptical track segments.