Right cones

[mikah]

So you have a right cone with a certain opening angle, shouldn't there be an equation to find the height for a certain diameter? Seeing as the sides move away at a constant rate there should be some sort of equations realting angle to diameter.

[questionposter]

So you have a right cone with a certain opening angle, shouldn't there be an equation to find the height for a certain diameter? Seeing as the sides move away at a constant rate there should be some sort of equations realting angle to diameter.

 

Which part of a cross-section of a cone are you defining as the hypotenuse of the triangle? Because you can use the side as either a slant height or the hypotenuse. The general way to find the volume or surface area of a cone already uses trigonometry, or just 1/3 that of the cylinder it would make, so if there's some kind of angle you'd probably just use a trigonometric function to find a side length and that's about it.

Depending on how you arrange the equation for a cone you can get the different parts of it.

Edited October 13, 2011 by questionposter

[mikah]

I never said anything about a cross-section or volume or surface area, and the side is the slant, also I'm not sure where you're pulling triangles from, since we're working in three-dimensions. I'm saying if you have a whole cone, a standard right cone, the sides should move away at a specific rate for every angle the vertex should be, so with that knowledge there should be an equation to find your height using only diameter and opening angle.

[questionposter]

I never said anything about a cross-section or volume or surface area, and the side is the slant, also I'm not sure where you're pulling triangles from, since we're working in three-dimensions. I'm saying if you have a whole cone, a standard right cone, the sides should move away at a specific rate for every angle the vertex should be, so with that knowledge there should be an equation to find your height using only diameter and opening angle.

 

I don't know how much geometry you've taken, but it seems like your only starting to learn geometry. You can still easily use 2D objects to calculate parts of a 3D object. The height is a line right? Well that's just one dimension, and what does a triangle have? 2. It's made up off lines. So what you do is you draw a right triangle inside the cone which consists of the slant height as the hypotenuse, the radius of the base as a leg, and the actual height as the other leg, and in order to get the height you need to know the slant height or radius along with an angle, or use Pythagorean theorem if you know two of the 3 sides of the triangle. So if you know the radius, here's what you do:

 

In an advanced calculator like a scientific or graphing one, you can type "tan" and in the parentheses put the angle which is between the hypotenuse and the radius of the base.

 

 

tan(angle)=height/radius

 

and then re-arrange the equation to make

 

tan(angle)*radius=height

 

 

And if you know the angle and slant height which is the same as the hypotenuse

 

sin(angle)=height/hypotenuse

 

then re-arrange it to make

 

sin(angle)*hypotenuse=height

 

and there you go.

 

And there is definitely a relationship between the slant height and the actual height and it's because a cone does have inherent triangular properties about it. The only way they could be related is because of this triangle thing I'm talking about, because the height and slant height are connected at the same point, and can be connected by a straight line after a specific distance which will form a certain set of angles.

Edited October 14, 2011 by questionposter