Surface formula of elliptical cone
- Thread starter niksirat
- Start date
D
Deleted member 4993
Guest
Can you derive the formula for surface area of a "circular" cone?
Please show us what you have tried and exactly where you are stuck.
Please follow the rules of posting in this forum, as enunciated at:
Please share your work/thoughts about this problem
Dr.Peterson
Elite Member
- Joined
- Nov 12, 2017
- Messages
- 16,092
By "degree" do you mean "angle"?
But you show v in two incompatible places; please clarify how you are defining it.
The surface area formula (in terms of a, b, and h) requires an elliptic integral:
Elliptic Cone -- from Wolfram MathWorld
A cone with elliptical cross section. The parametric equations for an elliptic cone of height h, semimajor axis a, and semiminor axis b are x = a(h-u)/hcosv (1) y = b(h-u)/hsinv (2) z = u, (3) where v in [0,2pi) and u in [0,h]. The elliptic cone is a quadratic ruled surface, and has volume...
mathworld.wolfram.com
However you are defining v, you will need to determine h in terms of a, b, and v. Have you tried that?
Elliptical cone formula is
[math]z^2/h^2=x^2/a^2+y^2/b^2[/math]I want to write it with V parameter.
Dr.Peterson
Elite Member
- Joined
- Nov 12, 2017
- Messages
- 16,092
Ah! When you said "surface formula", you didn't mean surface area, as I assumed following SK, but the equation of the surface.
But you haven't clarified what V is. Is it the angle from the base to the vertex at a particular point? The angle will vary.
Once you specify that, if it is to be measured at a vertex or co-vertex, then you just need basic trigonometry to find h.
blamocur
Elite Member
- Joined
- Oct 30, 2021
- Messages
- 2,627
As previous posters pointed out, the answer depends on the place where you measure your angle. Once you decide on that you can express [imath]h[/imath] in terms of [imath]V[/imath] and [imath]a[/imath] or [imath]b[/imath], or both.