Theory question 106
Mensuration of Plane Shapes(a) In the diagram below, O is the centre of the circle radius 3.2cm. If ∠PRQ = 42°, calculate, correct to two decimal places, the area of the: (i) minor sector POQ; (ii) shaded part (b) if the sector POQ in (a) is used to form the curved surface of a cone with vertex O, calculate the base radius of the cone, correct to one decimal place.
Show the model answer
Model answer
∠POQ = 2 × 42° = 84° (angle at centre = twice angle at circumference).
(a)(i) Sector area = 84/360 × 22/7 × 3.22 = 7.51 cm².
(ii) Area of △POQ = ½ × 3.22 × sin 84° = 5.12 × 0.9945 = 5.09 cm².
Shaded segment = 7.51 − 5.09 = 2.42 cm².
(b) Arc PQ = 84/360 × 2π × 3.2 = 4.69 cm = circumference of the base.
2πr = 4.69 ⇒ r = 84/360 × 3.2 = 0.75 ≈ 0.7 cm (1 d.p.).
Explanation
The shaded segment is the sector minus the triangle.