Sector of a Circle: Area, Arc Length, Perimeter

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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