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.
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.