Sectors and segments appear in Class 10 'Areas related to circles'. Enter the radius and the angle at the centre to get the arc length, the sector area and the segment area (the minor segment for angles up to 180°).
Enter the radius.
Enter the central angle in degrees.
Read the arc length, sector and segment areas.
Arc length: L = (θ ÷ 360) × 2πr
Sector area: A = (θ ÷ 360) × πr²
When two radii cut a circle, they mark off three related pieces: the arc, which is the curved bit of the boundary between them; the sector, which is the pizza-slice region between the radii and the arc; and the segment, which is the region between the arc and the straight chord joining its ends. Given the radius and the angle at the centre in degrees, this calculator returns the arc length, the sector area and the segment area.
These shapes are the core of the Class 10 chapter 'Areas related to circles'. Outside the classroom they describe the sweep of a car wiper, the area watered by a rotating sprinkler, the path of a pendulum, the length of a curved road or railway bend, and the portion of a pie chart.
1. Note the radius r and the central angle θ in degrees.
2. Find the fraction of the full circle: θ ÷ 360.
3. Arc length: L = (θ ÷ 360) × 2πr.
4. Sector area: A = (θ ÷ 360) × πr².
5. Convert θ to radians for the segment: θ (rad) = θ × π ÷ 180.
6. Segment area: sector area minus the triangle formed by the two radii, which is ½ r² (θ − sin θ) with θ in radians.
7. For the major sector or major arc, subtract the minor one from the full circle.
A central angle of θ degrees takes θ ÷ 360 of the full turn. Arc length and sector area are both proportional to that angle, because the circle is perfectly symmetric about its centre. So the arc is that fraction of the circumference 2πr, and the sector is that fraction of the area πr². A 90° sector is a quarter of the circle, a 60° sector a sixth. This proportional reasoning is all the formulas contain, and it is worth remembering in that form rather than as two separate rules.
A radian is defined so that an arc equal in length to the radius subtends one radian at the centre. The full circle is therefore 2π radians. In radians the arc length is simply L = rθ and the sector area is A = ½r²θ, with no 360 or π in sight. Note that the sector area is also ½ × L × r, just like the area of a triangle with base L and height r. Calculus and physics use radians for exactly this reason.
The triangle formed by the two radii and the chord has two sides of length r with the angle θ between them, so its area is ½r² sin θ. Taking it away from the sector leaves the segment, ½r²(θ − sin θ). The same formula still works when θ is more than 180°: sin θ turns negative, so the triangle is effectively added, giving the major segment. At θ = 180° the chord is a diameter and the segment equals the sector, a semicircle.
Arjun's car has a wiper blade 25 cm long that sweeps through an angle of 115°. He wants the length of the curve traced by the blade tip and the area of glass one blade cleans.
Arc length = (θ ÷ 360) × 2πr: = (115 ÷ 360) × 2π × 25 = 50.1782
Sector area = (θ ÷ 360) × πr²: = (115 ÷ 360) × π × 25² = 627.2277
Segment area = sector − triangle = ½r²(θ − sin θ): = 344.0065
Answer: Arc length 50.1782; Sector area 627.2277; Segment area 344.0065
Using sin θ with θ in degrees inside the segment formula, which needs radians.
Using the diameter instead of the radius.
Confusing the sector with the segment when a question asks for the region cut off by a chord.
Forgetting to subtract from the whole circle when the question asks for the major sector.
Rounding π as 3.14 early and then comparing with a textbook answer that uses 22/7.
Class 10 problems on wipers, sprinklers, clock hands, brooches and flower beds.
Finding the length of a curved road, track or railway bend of known radius.
Estimating the area covered by a rotating sprinkler or a lighthouse beam.
Cutting fabric or sheet for fans, lampshade patterns and cones.
Drawing pie charts, where each slice is a sector proportional to its share.
What is the major sector?
Subtract the minor sector from the full circle area πr².
Which π is used?
The full-precision value; answers using 22/7 will differ slightly.