Theory question 103
Mensuration of SolidsA sector of a circle with radius 21 cm has an area of 280 cm2. (a) Calculate, correct to 1 decimal place, the perimeter of the sector. (b) If the sector is bent such that its straight edges coincide to form a cone, calculate, correct to the nearest degree, the vertical angle of the cone. [Take π = 22/7]
Show the model answer
Model answer
(a) Area = ½ × r × arc, so arc = 2 × 280/21 = 26.67 cm (the angle is about 72.7°).
Perimeter = 26.67 + 21 + 21 ≈ 68.7 cm.
(b) The arc becomes the base circumference: 2 × 22/7 × r = 26.67, so r ≈ 4.24 cm. The slant height is 21 cm.
sin(half the vertical angle) = 4.24/21 ≈ 0.202, so half-angle ≈ 11.7°.
Vertical angle ≈ 23°.
Explanation
The radius of the sector becomes the slant height of the cone.