Question

Find the volume of a cone formed from a sector of diameter 72 inches and a central angle 7pi/6.

Answer 1

@~Gelmhar
If the diameter of the sector is 72 inches, then its radius, which is also the slant height of the cone, is 36 inches. The circumference of that complete circle would have been C = πd or C = 72π.

If the sector had a central angle of 7π/6, then the length of its arc is (7π/6)/(2π) * 72π = (7/12) * 72π = 42π

42π is also the circumference of the base of the cone to be formed.

If 42π is the circumference of the base of the cone, then its diameter is 42, so its radius is 21 inches.

If its radius is 21 inches and its slant height is 36 inches, then find the cone's height:

a² + b² = c²

radius² + cone's height² = slant height²

21² + h² = 36²

441 + h² = 1296

h² = 855

h = 3√95 inches


volume = ⅓ area of base x height

V = ⅓ πr²h

V = ⅓ π(21)²(3√95)

V = 441π√95 <==ANSWER

V = 13,603.62 in³ <==decimal ANSWER

I hope that helps!! :-)