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1. Surface area of a cone
I want to prove that the surface area of a cone of radius r and height h is where l is the lateral height of the cone.

We take a ring of width distance from its apex.

The surface area of the ring is where y is the radius of the ring.

Using similar triangles, therefore, surface area is

Adding all of these rings together,

What am I doing wrong?
2. Re: Surface area of a cone
(Original post by r2enigma)
I want to prove that the surface area of a cone of radius r and height h is where l is the lateral height of the cone.

We take a ring of width distance from its apex.

The surface area of the ring is where y is the radius of the ring.

Using similar triangles, therefore, surface area is

Adding all of these rings together,

What am I doing wrong?
It's easier to consider its net.

Are you required to use integration?
3. Re: Surface area of a cone
(Original post by r2enigma)
I want to prove that the surface area of a cone of radius r and height h is where l is the lateral height of the cone.

We take a ring of width distance from its apex.

The surface area of the ring is where y is the radius of the ring.

Using similar triangles, therefore, surface area is

Adding all of these rings together,

What am I doing wrong?
You've assumed that dx is an infinitesimal increase in x but it's not because x is the distance from the apex and dx is the distance along the cone's surface not the axis.
4. Re: Surface area of a cone
(Original post by r2enigma)
I want to prove that the surface area of a cone of radius r and height h is where l is the lateral height of the cone.

We take a ring of width distance from its apex.

The surface area of the ring is where y is the radius of the ring.
This is wrong. A cone is made up of "rings" with sloping sides, and so their surface area is bigger than 2 pi y dx (which would be correct if the slides did NOT slope).

[I think. It's not 100% clear what you're doing without diagrams].
5. Re: Surface area of a cone
ahhh yes. now I get it. thanks guys