A sector is a slice of a circle formed by two radii and the arc between them, like one piece of a pizza. A circular segment is the region between a chord and its arc, like a curved cap cut off from the circle. These areas matter in geometry, engineering, design, architecture, and any situation where circular parts are divided or removed.
The key idea is that a central angle controls what fraction of the full circle is included.
Understanding Geometry: Sector and Segment Area
The central angle does more than name the opening. It tells you how much of the circle has been swept out as two radii rotate from one position to another. A small angle produces a narrow region.
An angle of one hundred eighty degrees produces half of the circle. This proportional idea is useful because the radius stays fixed while the angle changes. If the angle doubles, the sector area doubles.
Its arc length doubles too. The relation is linear in the angle, not in the visible width of the curved edge. A sector with a larger radius can still have a shorter arc than one with a smaller radius if its angle is much smaller.
A segment needs an extra step because its straight boundary is a chord rather than two radii. Draw both radii from the centre to the chord endpoints. This creates a triangle inside the sector.
The curved cap is what remains after that triangle is removed. This works for a minor segment, the smaller cap made by an angle less than one hundred eighty degrees. For a major segment, use the whole circle area minus the minor segment area.
A diagram matters here. Students often subtract the wrong region because they do not first mark which side of the chord is wanted.
The triangle calculation depends on the angle between the radii. Its area is found from the two equal side lengths and the sine of the included angle. Sine is especially helpful because the chord is often unknown or inconvenient to measure.
Check the mode on a calculator before using sine. A problem written with degrees needs degree mode. A problem written with radians needs radian mode.
Radians are not a different kind of angle shape. They measure the same turn in a different way, based on arc length compared with radius. This is why formulas involving radians often look shorter.
These shapes appear when a round object is cut, filled, or covered only partly. Examples include a curved window panel, a fan blade outline, a circular pond edge, a gauge face, and material removed from a metal disc. Area answers must use square units, such as square centimetres.
Arc length uses ordinary length units, such as centimetres. Keep those units separate. A useful reasonableness check is to compare the result with the full circle.
A minor sector must be smaller than the circle. A minor segment must be smaller than its sector. For a very small angle, the segment should be extremely thin because the chord lies close to the arc.
Key Facts
- Full circle area: A = πr^2
- Sector area in degrees: Asector = (θ/360)πr^2
- Arc length in degrees: s = (θ/360)2πr
- Triangle area with two radii: Atriangle = (1/2)r^2 sin θ, where θ is in degrees for calculator mode
- Minor segment area: Asegment = Asector - Atriangle
- For radians, sector area is Asector = (1/2)r^2θ
Vocabulary
- Sector
- A sector is the region of a circle enclosed by two radii and the arc between them.
- Circular segment
- A circular segment is the region of a circle enclosed by a chord and the arc between the chord's endpoints.
- Central angle
- A central angle is an angle whose vertex is at the center of a circle and whose sides are radii.
- Chord
- A chord is a line segment with both endpoints on a circle.
- Arc
- An arc is a connected part of the circumference of a circle.
Common Mistakes to Avoid
- Using θ/180 instead of θ/360 for sector area is wrong because the angle must be compared to the full 360 degrees of a circle.
- Subtracting the sector from the triangle for a minor segment is wrong because the curved segment is the sector area minus the isosceles triangle area.
- Forgetting to square the radius in A = πr^2 gives the wrong units and makes the area too small or too large.
- Mixing degrees and radians in formulas causes incorrect results because Asector = (θ/360)πr^2 uses degrees while Asector = (1/2)r^2θ uses radians.
Practice Questions
- 1 A circle has radius 8 cm and central angle 90 degrees. Find the area of the sector in terms of π and as a decimal using π ≈ 3.14.
- 2 A circle has radius 10 m and central angle 60 degrees. Find the area of the minor circular segment using Asegment = Asector - Atriangle and sin 60 degrees ≈ 0.866.
- 3 A chord cuts off a small cap from a circle. Explain why the area of that cap is found by subtracting a triangle from a sector rather than by using only the chord length.