The chord AB of a circle with centre at O is 2\sqrt{3} times the height of the minor segment. If P is the area of the sector OAB and Q is the area of the minor segment of the circle, then what is the approximate value of \frac{P}{Q}? (Take \sqrt{3}=1.7 and \pi=3.14)

Explanation

Let central angle be 2\theta. Chord = 2R \sin \theta. Height of segment = R - R\cos \theta. Given 2R \sin \theta = 2\sqrt{3}R(1 - \cos \theta), which simplifies to \tan(\theta/2) = 1/\sqrt{3} \implies \theta=60^{\circ}. So central angle is 120^{\circ}. P = \frac{1}{3}\pi R^2, Q = R^2(\frac{\pi}{3} - \frac{\sqrt{3}}{4}). \frac{P}{Q} = \frac{4\pi}{4\pi - 3\sqrt{3}} = \frac{12.56}{12.56 - 5.1} \approx 1.68 \approx 1.7.

Related questions on Mensuration

• A pendulum swings through an angle of 30° and its end describes an arc of length 55 cm. What is the length of the pendulum? (Take $\pi = 22/...
• A conical tent has an angle of 60° at the vertex. If the curved surface area is 100 m^2, then what is the volume of the tent?
• A right circular cone and a hemisphere have equal base and equal volume. What is the ratio of the height of the cone to the radius of the he...
• A wire is in the form of an equilateral triangle with an area of 36\sqrt{3} cm^2. If it is changed into a semicircle, then what is its r...
• Let the area of the largest possible square inscribed in a circle of unit radius be x. Let the area of the largest possible circle inscribed...