Sector Area Formula

The Formula

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° each), each slice has \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 6 and central angle 60°: A = \frac{60}{360} \cdot \pi(6)^2 = \frac{1}{6} \cdot 36\pi = 6\pi \approx 18.85 \text{ square units}

Notation

A for area, r for radius, \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° each), each slice has \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 = \frac{1}{2}r^2\theta for \theta in radians; equivalently A = \frac{\theta}{2\pi} \cdot \pi r^2; in polar coordinates: A = \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 8 cm and central angle 90°.

Solution

  1. 1
    Step 1: Write the sector area formula in degrees: A = \frac{\theta}{360°} \times \pi r^2.
  2. 2
    Step 2: Substitute \theta = 90° and r = 8 cm: A = \frac{90}{360} \times \pi (8)^2.
  3. 3
    Step 3: Simplify the fraction: \frac{90}{360} = \frac{1}{4}, and r^2 = 64.
  4. 4
    Step 4: Compute: A = \frac{1}{4} \times 64\pi = 16\pi \approx 50.27 cm².

Answer

A = 16\pi \approx 50.27 cm²
A 90° sector is one-quarter of the full circle. One-quarter of the circle's area \pi(8)^2 = 64\pi cm² gives 16\pi cm². This matches \frac{1}{4}\pi r^2 for a quarter-circle.

Example 2

medium
A sector has a central angle of \frac{2\pi}{3} radians and a radius of 9 cm. Find its area.

Common Mistakes

  • Using degrees in the radian formula without converting
  • Confusing sector area with the area of the entire circle
  • Mixing up sector area (\frac{\theta}{360} \cdot \pi r^2) with arc length (\frac{\theta}{360} \cdot 2\pi r)

Why This Formula Matters

Used in data visualization (pie charts), engineering (fan blades, windshield wipers), and calculating areas of irregular regions involving circles.

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° each), each slice has \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?

A for area, r for radius, \theta for central angle

Why is the Sector Area formula important in Math?

Used in data visualization (pie charts), engineering (fan blades, windshield wipers), and calculating areas of irregular regions involving circles.

What do students get wrong about Sector Area?

Like arc length, make sure the angle units match the formula. The radian form (\frac{1}{2}r^2\theta) is simpler for calculus applications.

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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