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Radians, arcs and sectors
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Radians measure angles using the radius of a circle. They make the formulae for arcs and sectors very simple.
What is a radian?
One radian is the angle at the centre when the arc length equals the radius.
π radians = 180°
90° = π2, 60° = π3, 45° = π4, 30° = π6
To convert radians to degrees multiply by 180π.
π radians = 180°
90° = π2, 60° = π3, 45° = π4, 30° = π6
To convert radians to degrees multiply by 180π.
Arc and sector
With θ in radians:
Arc length s = rθ
Sector area A = 12r2θ
r = 8, θ = 1.2: s = 9.6
Arc length s = rθ
Sector area A = 12r2θ
r = 8, θ = 1.2: s = 9.6
Segment
A segment is a sector minus the triangle.
Segment area = 12r2θ − 12r2 sin θ = 12r2(θ − sin θ)
Segment area = 12r2θ − 12r2 sin θ = 12r2(θ − sin θ)
A sector has radius 5 cm and angle 1.4 radians. Find its perimeter.
- Arc: 5 × 1.4 = 7
- Two radii: 10
Answer: 17 cm
π radians = 180°. s = rθ and A = 12r2θ with θ in radians.
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