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# Sector of a Circle: Area, Arc Length, Perimeter

## What Is a Sector of a Circle?

A **sector of a circle** is the region bounded by two radii and the arc between them, the pie-slice or pizza-slice shape. The angle between the two radii, measured at the centre, is the sector's **central angle**, written θ.

Every sector comes in a pair. The smaller slice, with a central angle less than 180°, is the **minor sector**; the larger piece, with an angle greater than 180°, is the **major sector**. Together they make the whole circle. A sector is easy to confuse with a segment: a sector is bounded by two straight radii and an arc, while a segment is bounded by a straight chord and an arc. Two radii means sector; one chord means segment.

## The Sector Formulas — Area, Arc Length, and Perimeter

Every sector formula is the same single idea: a sector is the fraction θ/360° of the whole circle (in degrees), so it takes that same fraction of the circle's area and of its circumference. Define the variables once: **r** is the radius, **θ** the central angle, **l** the arc length (the curved edge of the sector), and **π** ≈ 3.14159.

### Area of a sector:

The whole circle's area is πr², and the sector is the angle's fraction of it:

Area=θ/360°×πr² (θ in degrees).

In radians, a full circle is 2π, so the fraction becomes θ/2π, and the area simplifies neatly:

Area=1/2 r²θ (θ in radians).

### Arc length:

The arc is the same fraction of the full circumference 2πr:

l=θ/360°×2πr (θ in degrees), l=rθ (θ in radians).

### Area without the angle:

If you know the arc length l and radius r but not the angle, the area is:

Area=1/2 l r.

**Perimeter of a sector.** The boundary of a sector is two straight radii plus the curved arc, so:

Perimeter=2r+l.

## Examples of the Sector of a Circle

### Example 1 - A sector of a circle has a radius of 6 cm and a central angle of 60°. Find its area. Use π=3.14.

The sector is 60/360=1/6 of the circle:

Area=θ/360°×πr²=60/360×3.14×6²=1/6×113.04=18.84 cm².

Final answer: 18.84 cm².

### Example 2 - A sector has a radius of 7 cm and an arc length of 11 cm. Find its area. Use π=22/7.

The correct formula for the area from the arc length is 1/2 l r:

Area=1/2×11×7=38.5 cm².

Final answer: 38.5 cm².

### Example 3 - A sector has a radius of 5 m and a central angle of 2 radians. Find its area and arc length.

Area=1/2×5²×2=25 m², l=rθ=5×2=10 m.

Final answer: area 25 m², arc length 10 m.

### Example 4 - Find the perimeter of a sector of radius 10 cm whose central angle is 72°. Use π=3.14.

l=72/360×2×3.14×10=12.56 cm.

Perimeter=2r+l=32.56 cm.

Final answer: 32.56 cm.

### Example 5 - A sector of a circle of radius 12 cm has an area of 48π cm². Find its central angle in degrees.

θ/360°×π×12²=48π; θ=48/144×360°=120°.

Final answer: θ=120°.

### Example 6 - A windscreen wiper of length 25 cm sweeps through an angle of 108°. Find the area of the windscreen it cleans. Use π=3.14.

Area=108/360×3.14×25²=588.75 cm².

Final answer: 588.75 cm².

## Common Errors When Working With Sectors

### Mistake 1: Forgetting the one-half in the arc-length area formula

**Correct way:** The area from the arc length is 1/2 l r.

### Mistake 2: Mixing degrees and radians in the same formula

**Correct way:** Match the formula to the unit. Degrees use θ/360°×πr²; radians use 1/2 r²θ.

### Mistake 3: Confusing a sector with a segment

**Correct way:** A sector is bounded by two radii and an arc; a segment is bounded by a chord and an arc.

## Key Takeaways

- A **sector of a circle** is the region between two radii and an arc; the smaller is the minor sector, the larger the major sector.
- A sector's area is the angle's fraction of the whole circle: θ/360°×πr² in degrees, or 1/2 r²θ in radians.
- The arc length is θ/360°×2πr (degrees) or rθ (radians), and the perimeter is 2r+l.
- A sector is bounded by two radii; a segment is bounded by a chord, so segment area = sector area − triangle area.