For a cone with radius 3 cm and slant 5 cm, what is the volume and total surface area?
Given the radius of a cone is r= 3 cm
The slant s = 5 cm
We need to find the volume and the surface area.
First we will calculate the surface area.
The surface area of the cone = surface area of the base + surface area of the slant.
==> Surface area of the base = r^2 * pi = 3^2 pi = 9pi
==> surface area of the slant = r*s*pi 3*5*pi = 15pi.
==> Then the total surface area = 15pi + 9pi = 24pi
Now we will determine the volume.
We know that :
V = (1/3) r^2 * pi * h
==> We will calculate the height h.
==> h= sqrt( s^2 -r^2) = sqrt(25-9) = sqrt16 = 4
==> h= 4
==> V = (1/3)*r^2 * h * pi
= (1/3)* 9 * 4 * pi = 12pi.
Then the volume of the cone is 12pi cm^3 and the surface area is 24pi cm^3.
The total surface area of a cone is the sum of the area of the base and that of the curved surface.
The area of the base is pi*r^2. Here r = 3 cm
So the area of the base is pi*9
The area of the curved surface area is pi*r*s, here r = 3 and s = 5
The curved surface area is pi*15
The total surface area = 15*pi + 9*pi = 24*pi
The volume of a cone is (1/3)*pi*r^2*h
Here h = sqrt (5^2 - 3^2)
=> sqrt (25 - 9)
=> sqrt 16
Volume = (1/3)*pi*9*4
The required total surface area = 24*pi square units and the volume is 12*pi cubic units