Answer:
The area of the segment AYB is 271.0413985 cm²
Step-by-step explanation:
The rule of the area of a sector is A = [tex]\frac{\alpha }{360}[/tex] × π r², where
- α is the central angle of the sector
- r is the radius of the circle
The area of a triangle is A = [tex]\frac{1}{2}[/tex] × s1 × s2 × sinФ
- s1 and s2 are two sides of the triangle
- Ф is the included angle between s1 and s2
∵ Area of the segment AYB = Area the sector AOB - Area ΔAOB
∵ The radius of the circle is 21 cm
∴ r = 21 cm
∵ The central angle of the sector is 120°
∴ α = 120°
∵ π = [tex]\frac{22}{7}[/tex]
→ Substitute them in the rule of the area of the sector to find it
∵ Area sector AOB = [tex]\frac{120}{360}[/tex] × [tex]\frac{22}{7}[/tex] × (21)²
∴ Area of sector AOB = 462 cm²
∵ OA and OB are the radii of the circle
∴ s1 = OA and s2 = OB
∴ s1 = s2 = 21 cm
∵ The angle included between them is 120°
∴ Ф = 120°
→ Substitute them in the rule of the area of the triangle to find it
∵ Area of the Δ = [tex]\frac{1}{2}[/tex] × 21 × 21 × sin(120)
∴ Area of the Δ = 110.25√3 cm²
∵ Area of the segment AYB = Area the sector AOB - Area ΔAOB
∴ Area of the segment AYB = 462 - 110.25√3
∴ Area of the segment AYB = 271.0413985 cm²
