Central angle
A circle k with a center at point S and a radius of 6 cm is given. Calculate the size of the central angle subtended by a chord 10 cm long.
Correct answer:
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Dr. Math
1. Use the chord length formula:
The length L of a chord subtending a central angle θ (in radians) in a circle of radius r is:
Plugging in L = 10 cm and r = 6 cm:
2. Solve for θ :
In degrees:
The length L of a chord subtending a central angle θ (in radians) in a circle of radius r is:
L = 2r sin(θ2)
Plugging in L = 10 cm and r = 6 cm:
10 = 2 × 6 × sin(θ2) ⇒ sin(θ2) = 1012 = 56.
2. Solve for θ :
θ2 = arcsin(56) ⇒ θ = 2 arcsin(56).
In degrees:
θ ≈ 2 × 56.44° ≈ 112.89°.
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geometryalgebraplanimetricsgoniometry and trigonometryUnits of physical quantitiesGrade of the word problem
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