Sector Area Calculator

Calculate the area of a circle sector

Radius (r)

Angle (θ)

Step 1: Identify the Radius

Measure or determine the distance from the center to the edge of the sector.

Why: The radius is fundamental to calculating area using the quadratic relationship (A ∝ r²).

Step 2: Determine the Central Angle

Find the angle at the center of the circle that bounds the sector.

Why: The angle determines what fraction of the full circle your sector represents.

Step 3: Select the Angle Unit

Choose whether to work with degrees (0-360°) or radians (0-2π).

Why: Different formulas apply based on the unit; radians simplify the formula to A = ½r²θ.

Step 4: Apply the Corresponding Formula

Use (θ/360) × πr² for degrees or ½ × r² × θ for radians.

Why: Each formula accounts for the proportional relationship between the angle and the full rotation (360° or 2π radians).

Step 5: Verify and Interpret Results

Check that the sector area is always less than the full circle area (πr²).

Why: A sector is always a portion of the circle, so it must be proportionally smaller than the full area.

Scenario: Calculate the area of a pizza slice with radius 10 cm and central angle 72° (one-fifth of a full circle).

Step 1 - Identify Radius: r = 10 cm (distance from center to edge of pizza).

Step 2 - Determine Angle: θ = 72° (the angle at the center of the pizza slice).

Step 3 - Select Unit: Working in degrees since problem uses 72°.

Step 4 - Apply Formula: A = (72/360) × π × 10² = (1/5) × π × 100 = 20π ≈ 62.83 cm².

Step 5 - Verify: Full circle area = π × 10² = 100π ≈ 314.16 cm². Sector (62.83) is one-fifth, confirming correctness.

Verification: 72° ÷ 360° = 0.2 = 1/5 ✓

Result: The pizza slice area is approximately 62.83 cm².

Interpretation: A 72° sector represents exactly 1/5 of the circle; scaling radius by 10 increases area by factor of 100.

The area of a sector depends on the angle unit used:

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