Arcs and Sectors: Arc Length, Sector Area and Segments

Slices of a Circle

A slice of pizza, the face of a hand fan, the path swept by a car's windscreen wiper, and the curved edge of an arched window are all parts of a circle. To find their lengths and areas, you need the formulas for the arc, sector and segment of a circle. These appear every year in WAEC and NECO mathematics, often combined with a cone made by folding a sector. This tutorial covers arcs and sectors step by step: the parts of a circle, arc length, the perimeter and area of a sector, finding the radius or angle, the same formulas in radians, chords and the area of a segment, and forming a cone from a sector, with worked examples. Unless stated otherwise, use π = 22/7.

Parts of a Circle

An arc is part of the circumference. The smaller one between two points is the minor arc, and the larger is the major arc.
A sector is the region bounded by two radii and an arc, shaped like a slice of pizza.
A chord is a straight line joining two points on the circumference.
A segment is the region between a chord and an arc.
The angle θ between the two radii, at the centre, is the angle subtended by the arc.

The whole idea is simple: a sector with angle θ is the fraction θ/360 of the whole circle. So its arc is θ/360 of the circumference, and its area is θ/360 of the circle's area.

Length of an Arc

The circumference of a circle is 2πr, so:

arc length = (θ/360) × 2πr

Worked example 1

Find the length of an arc of a circle of radius 14 cm that subtends an angle of 60° at the centre.

Arc length = (60/360) × 2 × 22/7 × 14 = (1/6) × 88 ≈ 14.67 cm.

Worked example 2: a clock

The minute hand of a clock is 7 cm long. How far does its tip travel in 20 minutes?

In 20 minutes the hand turns through 20/60 × 360° = 120°.

Distance = (120/360) × 2 × 22/7 × 7 = (1/3) × 44 ≈ 14.67 cm.

Area of a Sector

The area of a circle is πr², so:

area of sector = (θ/360) × πr²

Worked example 3

Find the area of a sector of radius 7 cm and angle 72°.

Area = (72/360) × 22/7 × 7² = 0.2 × 154 = 30.8 cm².

Perimeter of a Sector

The perimeter of a sector is the arc plus the two radii:

perimeter = arc length + 2r

Worked example 4

Find the perimeter of a sector of radius 10.5 cm and angle 120°.

Arc = (120/360) × 2 × 22/7 × 10.5 = (1/3) × 66 = 22 cm.

Perimeter = 22 + 2 × 10.5 = 43 cm. Forgetting the two radii is the most common mistake.

Finding the Angle or the Radius

Substitute what you know into the formula and solve for the unknown.

Worked example 5

An arc of length 11 cm lies on a circle of radius 7 cm. Find the angle it subtends at the centre.

11 = (θ/360) × 2 × 22/7 × 7 = (θ/360) × 44.

So θ/360 = 11/44 = 1/4, and θ = 90°.

Worked example 6

A sector of angle 90° has an area of 154 cm². Find the radius.

154 = (90/360) × 22/7 × r², so r² = 154 × 4 × 7/22 = 196, and r = 14 cm.

Arcs and Sectors in Radians

In university mathematics and further mathematics, angles are measured in radians, where 2π radians = 360°. The formulas then become very simple:

arc length s = rθ and sector area A = ½r²θ (θ in radians)

These follow from the degree formulas by replacing θ/360 with θ/2π.

Worked example 7

A sector has radius 5 cm and angle 1.2 radians. Find its arc length and area.

s = 5 × 1.2 = 6 cm, and A = ½ × 25 × 1.2 = 15 cm².

The tutorial on trigonometric ratios explains how to convert between degrees and radians.

Chords and the Area of a Segment

A chord AB subtends angle θ at the centre O. Triangle OAB is isosceles, with two sides equal to r. Useful results:

Length of the chord = 2r sin(θ/2), from splitting the isosceles triangle into two right-angled triangles.
Area of triangle OAB = ½r² sin θ.
Area of the minor segment = area of sector − area of triangle.

area of segment = (θ/360) × πr² − ½r² sin θ

Worked example 8

A chord subtends an angle of 60° at the centre of a circle of radius 10 cm. Find (a) the length of the chord and (b) the area of the minor segment. (Take π = 3.142.)

(a) Chord = 2 × 10 × sin 30° = 10 cm. (The triangle is equilateral, since θ = 60°.)

(b) Sector = (60/360) × 3.142 × 100 ≈ 52.37 cm². Triangle = ½ × 100 × sin 60° ≈ 43.30 cm².

Segment = 52.37 − 43.30 ≈ 9.07 cm².

The major segment is the rest of the circle: πr² minus the minor segment.

Forming a Cone from a Sector

When a sector is folded so that its two straight edges meet, it forms the curved surface of a cone. Two facts link the sector and the cone:

The radius of the sector becomes the slant height l of the cone.
The arc length of the sector becomes the circumference of the base of the cone: arc = 2πR, where R is the base radius.

Worked example 9

A sector of a circle of radius 21 cm with angle 120° is folded to form a cone. Find (a) the base radius of the cone and (b) its height.

(a) Arc = (120/360) × 2 × 22/7 × 21 = 44 cm. So 2 × 22/7 × R = 44, giving R = 7 cm.

(b) The slant height is 21 cm. By Pythagoras' theorem, h = √(21² − 7²) = √392 ≈ 19.8 cm.

The curved surface area of the cone, πRl = 22/7 × 7 × 21 = 462 cm², equals the area of the sector: (120/360) × 22/7 × 441 = 462 cm². This is a good check. The tutorial on mensuration covers cone volumes and surface areas.

Common Mistakes with Arcs and Sectors

Using the diameter instead of the radius. Halve the diameter first.
Leaving out the two radii when finding the perimeter of a sector.
Mixing degrees and radians. s = rθ and A = ½r²θ work only in radians.
Using the wrong value of π. Use 22/7 or 3.142 as the question instructs.
Confusing a sector with a segment. A segment is bounded by a chord, not by two radii.

Arcs and Sectors: Practice Questions

Find the length of an arc of radius 21 cm subtending 40° at the centre.
Find the area of a sector of radius 14 cm and angle 45°.
A sector of radius 14 cm has an area of 154 cm². Find its angle.
A sector has radius 6 cm and angle 2.5 radians. Find its arc length and area.
Find the area of the minor segment cut off by a chord that subtends 90° at the centre of a circle of radius 12 cm. (Take π = 3.142.)

Answers: 1. (40/360) × 2 × 22/7 × 21 ≈ 14.67 cm. 2. (45/360) × 22/7 × 196 = 77 cm². 3. 154 = (θ/360) × 616, so θ = 90°. 4. s = 15 cm, A = ½ × 36 × 2.5 = 45 cm². 5. Sector = ¼ × 3.142 × 144 ≈ 113.11 cm², triangle = ½ × 144 × sin 90° = 72 cm², so the segment ≈ 41.11 cm².

Conclusion

A sector of angle θ is the fraction θ/360 of a circle, so its arc length is (θ/360) × 2πr and its area is (θ/360) × πr². In radians these become s = rθ and A = ½r²θ. The perimeter of a sector adds the two radii, a segment is a sector minus the triangle ½r² sin θ, and when a sector is folded into a cone, its radius becomes the slant height and its arc becomes the base circumference. With these formulas, every arcs and sectors question becomes a short substitution.