(a)
In the diagram, XY is a chord of a circle of radius 5cm. The chord subtends an angle 96° at the centre. Calculate, correct to three significant figures, the area of the minor segment cut-off. (Take (pi = frac{22}{7})).
(b) The figure shows a circle inscribed in a square. If a portion of the circle is shaded with some portions of the square, calculate the total area of the shaded portions. [Take (pi = frac{22}{7})].
Explanation
(a) Area of minor sector = (frac{theta}{360} times pi r^{2})
= (frac{96}{360} times frac{22}{7} times 5^{2})
= (20.95 cm^{2})
Area of triangle formed from the sector = (frac{1}{2} r^{2} sin theta)
= (frac{1}{2} times 5^{2} times sin 96)
= (12.43 cm^{2})
(therefore text{The area of minor segment} = 20.95 – 12.43)
= (8.52 cm^{2})
(b) Area of minor sector = (frac{theta}{360} times pi r^{2})
= (frac{80}{360} times frac{22}{7} times 7^{2})
= (34.22 cm^{2})
Area of the square = ((14)^{2} = 196 cm^{2})
Area of the circle = (pi r^{2} = frac{22}{7} times 7^{2} = 154 cm^{2})
Area of the shaded portion in the square = (196 – 154 = 42 cm^{2})
Total area of the shaded portions = (42 + 34.22)
= (76.22 cm^{2} approxeq 76.2 cm^{2})