A sector of a circle of radius 12cm has the angle 120o. It is rolled up so that two bounding radii are joined to form a cone. Find the volume of the cone.
When a sector of a circle is rolled up in a given manner, we obtain a cone whose slant height is equal to the radius of the sector and the circumference if the base of the cone is equal to the length of the are of sector.
Now length of the acc of the sector square u
Let rein be the radius of the base, h cm be the height and l cm be the slant height of the cone. Then,
l = radius of the sector
= 12 cm.
Circumference of the base of the cone = Length of the arc of the sector
Now l2 = r2 + h2
Let V cm3 be the volume of the cone. Then,
Taking log on both sides logV
log22 + log128 + log2 - log21
logV = 2.2779
V = anti log (2.2779)
Hence the volume of the cone is 189.5 cm3.
Angle of the sector = 120Radius of the sector = 12 cm.Length of the arc of the sector = 2r= 2x 12= 8cmLet r be the radius of the cone formed after folding and joining the two radii. The circumference of the base of the cone is equal to the arc of the sector.2r = 8r = 4 cmThus, the radius of the cone = 4 cmlet h be the height of the conel2= r2+ h2h ==h2 == 128h == 8cmVolume of cone =r2h=x 4 x 4 x 8 = = 189.639 cm3
π(24)/3 = 8π
When the cone is rolled it will then have a circular base with this 8π circumference. The cone will also have a side length of 12 cm, which is important to find the height of the cone. First we need to find the radius of the base of the cone. We know that the circular base has a circumference of 8π and :
c=πd So we get:
d=8 and r=4 <--- Radius of the circular base.
For the height use the Pythagorean theorem:
Thus the formula for the volume of a cone:
V=1/3bh Where b is the area of the base.
V=1/3π128√2 <----- Answer
V=189.56 <----- Approximate answer