Sector Area Formula

Sector area is the area of a 'pie slice' region of a circle, bounded by two radii and the arc between them.

The Formula

A=12r2θ (radians)orA=θ360°⋅πr2 (degrees)A = \frac{1}{2}r^2\theta \text{ (radians)} \quad \text{or} \quad A = \frac{\theta}{360°} \cdot \pi r^2 \text{ (degrees)}

When to use: Imagine cutting a pizza into slices. Each slice is a sector. If you cut the pizza into 4 equal slices (90°90° each), each slice has 14\frac{1}{4} of the pizza's total area. The sector area is simply the fraction of the full circle determined by the central angle, applied to the total area.

Quick Example

A sector with radius 66 and central angle 60°60°: A=60360⋅π(6)2=16⋅36π=6π≈18.85 square unitsA = \frac{60}{360} \cdot \pi(6)^2 = \frac{1}{6} \cdot 36\pi = 6\pi \approx 18.85 \text{ square units}

Notation

AA for area, rr for radius, θ\theta for central angle

What This Formula Means

The area of a 'pie slice' region of a circle, bounded by two radii and the arc between them.

Imagine cutting a pizza into slices. Each slice is a sector. If you cut the pizza into 4 equal slices (90°90° each), each slice has 14\frac{1}{4} of the pizza's total area. The sector area is simply the fraction of the full circle determined by the central angle, applied to the total area.

Formal View

A=12r2θA = \frac{1}{2}r^2\theta for θ\theta in radians; equivalently A=θ2π⋅πr2A = \frac{\theta}{2\pi} \cdot \pi r^2; in polar coordinates: A=12∫θ1θ2r2 dθA = \frac{1}{2}\int_{\theta_1}^{\theta_2} r^2\,d\theta

Worked Examples

Example 1

easy
Find the area of a sector of a circle with radius 88 cm and central angle 90°90°.

Answer

A=16π≈50.27A = 16\pi \approx 50.27 cm²

First step

1
Step 1: Write the sector area formula in degrees: A=θ360°×πr2A = \frac{\theta}{360°} \times \pi r^2.

Full solution

  1. 2
    Step 2: Substitute θ=90°\theta = 90° and r=8r = 8 cm: A=90360×π(8)2A = \frac{90}{360} \times \pi (8)^2.
  2. 3
    Step 3: Simplify the fraction: 90360=14\frac{90}{360} = \frac{1}{4}, and r2=64r^2 = 64.
  3. 4
    Step 4: Compute: A=14×64π=16π≈50.27A = \frac{1}{4} \times 64\pi = 16\pi \approx 50.27 cm².
A 90° sector is one-quarter of the full circle. One-quarter of the circle's area π(8)2=64π\pi(8)^2 = 64\pi cm² gives 16π16\pi cm². This matches 14πr2\frac{1}{4}\pi r^2 for a quarter-circle.

Example 2

medium
A sector has a central angle of 2π3\frac{2\pi}{3} radians and a radius of 99 cm. Find its area.

Example 3

easy
Pizza of radius 99 is cut into 66 equal slices. Find the area of one slice in terms of π\pi.

Common Mistakes

  • Using 2πr2\pi r instead of πr2\pi r^2 — sector area scales the area, not the circumference.
  • Dropping the θ360°\frac{\theta}{360°} factor and reporting the whole circle's area — only the angle's share counts.
  • Treating the slice as a triangle — its outer boundary is a curved arc, so use the sector formula, not 12bh\frac{1}{2}bh.

Why This Formula Matters

It cements scaling a whole by an angle fraction in area units, the partner skill to arc length, and it is the bridge to integral area later; students who reach for 2πr2\pi r here are confusing perimeter with area. Recognizing it by "Am I asked for the area of a slice of the circle (square units), not the curved edge or the whole circle?" — rather than by familiar numbers — is what lets a student tell it apart from arc length and area of a circle and triangle area in a mixed problem set.

Frequently Asked Questions

What is the Sector Area formula?

The area of a 'pie slice' region of a circle, bounded by two radii and the arc between them.

How do you use the Sector Area formula?

Imagine cutting a pizza into slices. Each slice is a sector. If you cut the pizza into 4 equal slices (90°90° each), each slice has 14\frac{1}{4} of the pizza's total area. The sector area is simply the fraction of the full circle determined by the central angle, applied to the total area.

What do the symbols mean in the Sector Area formula?

AA for area, rr for radius, θ\theta for central angle

Why is the Sector Area formula important in Math?

It cements scaling a whole by an angle fraction in area units, the partner skill to arc length, and it is the bridge to integral area later; students who reach for 2πr2\pi r here are confusing perimeter with area. Recognizing it by "Am I asked for the area of a slice of the circle (square units), not the curved edge or the whole circle?" — rather than by familiar numbers — is what lets a student tell it apart from arc length and area of a circle and triangle area in a mixed problem set.

What do students get wrong about Sector Area?

The procedure for sector area is the easy part; the trap is using 2πr2\pi r instead of πr2\pi r^2. Asking "Am I asked for the area of a slice of the circle (square units), not the curved edge or the whole circle?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Sector Area formula?

Before studying the Sector Area formula, you should understand: area of circle, central angle.

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