Putting the formulas together
Real problems often combine several ideas from circular measure. The three key results are the arc length s = rθ, the sector area A = ½r2θ, and the segment area ½r2(θ − sin θ). The perimeter of a sector is the arc plus two radii, P = rθ + 2r, and the area of the triangle formed by two radii is ½r2sin θ. Choose the right combination for the shaded region in question.
Key formula
s = rθ · A = ½r2θ · segment = ½r2(θ − sin θ) · triangle = ½r2sin θ · sector perimeter = rθ + 2r
Working with composite regions
Shaded areas are usually found by adding or subtracting simple regions. A major sector equals the whole circle minus the minor sector, and a segment equals a sector minus a triangle. Sketch the figure, label each radius and angle, then decide what to add and what to subtract.
Worked example
A sector has radius 8 cm and angle 1.5 rad. Its arc length is 8 × 1.5 = 12 cm, and its area is ½ × 64 × 1.5 = 48 cm2. The perimeter of the sector is 12 + 2(8) = 28 cm. The area of the segment inside it is ½ × 64 × (1.5 − sin 1.5) = 32 × 0.5025 ≈ 16.08 cm2.
Angles in mixed units
If an angle is given in degrees, convert it to radians (× π/180) before using any of these formulas. Keep all lengths in the same unit and round only at the final step.
Remember
- Perimeter of a sector = arc length + 2r.
- Major sector = whole circle − minor sector.
- Segment = sector − triangle (½r2sin θ).