A circle can be divided into parts that have their own useful area formulas. A sector is the pie-slice region formed by two radii and an arc, while a segment is the curved region between a chord and an arc. These ideas matter in geometry, design, engineering, architecture, and any situation where circular shapes are cut into pieces.
Learning to identify the correct region is the first step toward using the correct formula.
Key Facts
- Sector area with degrees: A = (theta/360)pi r^2
- Sector area with radians: A = (1/2)r^2 theta
- Triangle area between two radii: A = (1/2)r^2 sin theta
- Minor segment area with radians: A = (1/2)r^2(theta - sin theta)
- Major segment area = pi r^2 - minor segment area
- Arc length with radians: s = r theta
Vocabulary
- Sector
- A sector is the region of a circle enclosed by two radii and the arc between them.
- Segment
- A segment is the region of a circle enclosed by a chord and the arc between the chord's endpoints.
- Chord
- A chord is a line segment whose endpoints both lie on the circle.
- Central angle
- A central angle is an angle with its vertex at the center of the circle and sides that are radii.
- Arc
- An arc is a connected part of a circle's circumference between two points.
Common Mistakes to Avoid
- Confusing a sector with a segment is wrong because a sector includes two radii, while a segment is bounded by a chord and an arc.
- Using the degree formula with a radian angle is wrong because A = (theta/360)pi r^2 requires theta in degrees, while A = (1/2)r^2 theta requires theta in radians.
- Finding only the sector area for a segment is wrong because a minor segment equals the sector area minus the isosceles triangle area.
- Using the diameter instead of the radius is wrong because the formulas use r, and the radius is half the diameter.
Practice Questions
- 1 A circle has radius 10 cm and central angle 72 degrees. Find the area of the sector in square centimeters, leaving your answer in terms of pi.
- 2 A circle has radius 6 m and central angle pi/3 radians. Find the area of the minor segment using A = (1/2)r^2(theta - sin theta). Give an exact answer.
- 3 A shaded region is bounded by two radii and the arc between their endpoints. Explain whether it is a sector or a segment, and state which area formula you would use first.