Question

A sector of a circle has radius 12 cm and an angle of 135. the sector is used to make a cone

Answer 1

When the sector is cut from the circle it will have a curved length of one third of the circle. This will be πd/3 where d is the 24 cm diameter of the circle. This yields: 

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: 

8π=πd 

d=8 and r=4 <--- Radius of the circular base. 

For the height use the Pythagorean theorem: 

r2+(height)2=(side )2

42+(height)2=122

h^2=144-16 

h2=128 

h=8√2 

Thus the formula for the volume of a cone: 

V=1/3bh Where b is the area of the base. 

V=1/3π(4)2(8√2) 

V=1/3π128√2 <----- Answer 

V=189.56 <----- Approximate answer