An inverted right cone, half filled with water. I've only the volume of the entire cone to go from.

Question

So the volume of the cone = 1/3 the base area * height

I've been told by a friend that the base area of a half scale cone = 1/4 of the larger cones base area; with the height being half.

So... That would make the water equal to the total volume * (Ab * 1/4) * (H * 1/2)

Vc = 20cm^3

Vw = Vc * 1/4 * 1/2
Vw = 20cm^3/8
Vw = 2.5cm^3

What I don't understand is why the base area of the scaled down cone = 1/4 of the base area of the larger cone.

Answer 1

suppose the radius of the original cone is r
then the base area of the original is πr^2

Now, the radius of your half-size cone is r/2
so the base area
= π(r/2)^2
= πr^2/4
= (1/4) πr^2

Thus the base area of the scaled down cone = 1/4 of the base area of the larger cone.