A cone is formed from a sector of a circular of radius 9cm which sustained an angel 120 find (a) the radius of the cone formed (b) the curved surface area of the cone (c) the total surface (d) volume
I used to illustrate this problem to my students by taking a cone-shaped drinking cup from a water dispenser and cutting it open to show just that kind of a sector.
It was easy to see that the arc of that sector becomes the circular base of the cone, so ....
since the angle is 120°, the arc length would be 1/3 of the circumference
(1/3)(2π)(9) cm = 6π cm
- this is the circumference of the base of the cone, and the slant height of the cone would be 9 cm
radius of cone ---- r
2πr = 6π
r = 3 cm
height^2 + 3^2 = 9^2
height^2 = 72
height = √72 = 6√2 cm
surface area of cone = πrs
= π(3)(9) = 27π cm^2
V = (1/3)π r^2
= (1/3)π(81) = 27π cm^3
check my arithmetic
To find the radius of the cone formed, we can use the formula:
r = (R * l) / s
where:
- r is the radius of the cone
- R is the radius of the circular sector (given as 9 cm)
- l is the slant height of the cone
- s is the length of the arc (which forms the sector)
To find the slant height (l) and the length of the arc (s), we need additional information. Could you provide the angle (θ) at the center of the circular sector?
Once we have the slant height, we can find the curved surface area, total surface area, and volume of the cone using the following formulas:
Curved Surface Area:
CSA = π * r * l
Total Surface Area:
TSA = π * r * (l + base radius)
Volume:
V = (1/3) * π * r^2 * h
where:
- CSA is the curved surface area
- TSA is the total surface area
- V is the volume
- π is a constant approximately equal to 3.14159
- h is the height of the cone
Please provide the angle at the center of the circular sector, and I will be able to calculate the cone's properties for you.