AJ Designer

Circle Sector Calculator

Area equals one half times radius squared times theta
rθKs

Solution

—
Share:

Sector Area

The sector area equals one-half times the radius squared times the central angle in radians.

A = ½r²θ

Radius from Sector Area

Determines the radius needed to produce a given sector area at a known central angle.

r = √(2A / θ)

Central Angle from Sector Area

Finds the central angle that produces a given sector area with a known radius.

θ = 2A / r²

How It Works

A sector is a "pie slice" of a circle bounded by two radii and the arc between them. The sector area formula A = ½r²θ uses the central angle in radians. If you know the angle in degrees, the calculator converts it automatically.

Example Problem

A sector has a radius of 10 cm and a central angle of 90°. Find the sector area.

  1. Identify the known values: radius r = 10 cm, central angle θ = 90°.
  2. Determine what to solve for: the sector area A.
  3. Convert 90° to radians: θ = 90 × π/180 = π/2 ≈ 1.5708 rad.
  4. Write the sector area formula: A = ½ × r² × θ.
  5. Substitute the values: A = ½ × 10² × 1.5708 = ½ × 100 × 1.5708.
  6. Compute the result: A ≈ 78.54 cm². This is one-quarter of the full circle area (π × 100 ≈ 314.16 cm²).

A 90° sector is exactly one-quarter of the circle, so its area is πr²/4.

When to Use Each Variable

  • Solve for Sector Area — when you know the radius and central angle, e.g., calculating the area of a pizza slice or pie chart segment.
  • Solve for Radius — when you know the sector area and angle, e.g., determining the radius needed for a circular garden bed of a given area.
  • Solve for Angle — when you know the sector area and radius, e.g., finding what fraction of a circle a given area represents.

Key Concepts

A sector is the region bounded by two radii and the arc between them — like a slice of pie. Its area is proportional to the central angle: A = ½r²θ (radians) or A = (θ/360)πr² (degrees). The sector area is always a fraction of the full circle area πr², where the fraction equals the central angle divided by a full revolution.

Applications

  • Data visualization: calculating the area of slices in pie charts and donut charts
  • Landscaping: determining material quantities for circular garden wedges and sprinkler coverage zones
  • Mechanical engineering: computing the cross-sectional area of sector-shaped ducts and channels
  • Land surveying: measuring areas of curved property boundaries defined by angular bearings

Common Mistakes

  • Forgetting to convert degrees to radians — the formula A = ½r²θ requires θ in radians
  • Confusing sector area with segment area — a sector includes the triangle from center to arc, while a segment is the region between the chord and the arc
  • Squaring the diameter instead of the radius — the formula uses r², not d²

Frequently Asked Questions

How do you calculate the area of a sector?

Use the formula A = ½r²θ, where r is the radius and θ is the central angle in radians. If the angle is in degrees, use A = (θ/360)πr². For a 10 cm radius and 90° angle, the sector area is (π × 100 × 90)/360 ≈ 78.54 cm².

What is the formula for sector area?

The sector area formula is A = ½r²θ (with θ in radians) or A = πr²θ/360 (with θ in degrees). Both give the same result. The formula derives from the fact that a sector's area is proportional to its central angle.

How do you find the area of a pizza slice?

Measure the radius of the pizza (half the diameter) and the angle of the slice. A pizza cut into 8 equal slices has 45° per slice. Then use A = πr²θ/360. For a 14-inch (35.6 cm) pizza with 8 slices: A = π × 17.8² × 45/360 ≈ 124.7 cm² per slice.

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

A sector is a 'pie slice' bounded by two radii and the arc between them. A segment is the region between a chord and the arc it cuts off. The segment area equals the sector area minus the triangle formed by the two radii and chord. Sectors always include the center point; segments never do.

What fraction of a circle is a 60-degree sector?

A 60-degree sector is 60/360 = 1/6 of the full circle. Its area is one-sixth of πr². Similarly, a 90° sector is 1/4, a 120° sector is 1/3, and a 180° sector (semicircle) is 1/2.

How do you convert degrees to radians for the sector formula?

Multiply the degree measure by π/180. For example, 90° × π/180 = π/2 ≈ 1.5708 radians. Common conversions: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π.

Can a sector area be larger than the circle area?

No. The maximum sector area equals the full circle area πr² when the central angle is 360° (2π radians). Any sector with a smaller angle will have a proportionally smaller area. Angles greater than 360° are not standard for sectors.

Sector Area Formula

The sector area formula calculates the area of a “pie slice” bounded by two radii and the arc between them:

A = ½ × r² × θ

Where:

  • A — sector area (in square units matching the radius)
  • r — radius of the circle
  • θ — central angle in radians (multiply degrees by π/180 to convert)

The sector area is proportional to the central angle. A full circle (θ = 2π) gives A = πr². A semicircle (θ = π) gives half that area.

Worked Examples

Land Surveying

What is the area of a pie-shaped lot with a 50 m radius and 72° angle?

A surveyor measures a wedge-shaped property plot radiating from a central point. The lot spans 72° with boundaries 50 m long.

  • Convert angle: 72° × π/180 = 1.2566 rad
  • A = ½ × 50² × 1.2566
  • A = ½ × 2500 × 1.2566
  • A ≈ 1,570.8 m²

This equals about 0.157 hectares or 0.388 acres. The sector formula is standard in cadastral surveying for radial subdivisions.

Sprinkler Design

How much lawn does a sprinkler cover if it sprays 8 m in a 120° arc?

A rotating sprinkler head covers a sector with an 8 m throw radius and 120° sweep angle. What area gets watered?

  • Convert angle: 120° × π/180 = 2.0944 rad
  • A = ½ × 8² × 2.0944
  • A = ½ × 64 × 2.0944
  • A ≈ 67.02 m²

Landscape designers use sector area to ensure full coverage with minimal overlap between adjacent sprinkler heads.

Data Visualization

What is the area of a 25% slice in a pie chart with radius 6 cm?

A pie chart is drawn with a 6 cm radius. A category representing 25% of the data occupies a 90° sector. What area does the slice cover?

  • Convert angle: 90° × π/180 = π/2 ≈ 1.5708 rad
  • A = ½ × 6² × 1.5708
  • A = ½ × 36 × 1.5708
  • A ≈ 28.27 cm²

This is exactly one-quarter of the full circle area (π × 36 ≈ 113.1 cm²), confirming the proportional relationship.

Related Calculators

Related Sites