# For a cone with radius 3 cm and slant 5 cm, what is the volume and total surface area?

hala718 | High School Teacher | (Level 1) Educator Emeritus

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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.

justaguide | College Teacher | (Level 2) Distinguished Educator

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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

=> 4

Volume = (1/3)*pi*9*4

=> 12*pi

The required total surface area = 24*pi square units and the volume is 12*pi cubic units