What is the formula for finding the surface area of an inclined cone?

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In summary, the area of an inclined cone (where the segment joining the tip and the center of the base circle is not perpendicular to the base plane) can be found by calculating the average of the surface areas of two right cones, one with a slant height equal to the shortest distance from the apex to the base and the other with a slant height equal to the greatest distance from the apex to the base. The surface of the cone can be parametrized and the tangent vectors and area can be calculated using integration.
  • #1
Aphex_Twin
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The area (not including the base) of a right cone is pi*radius*sqrt(height^2+radius^2).

What is the area of an inclined cone? (Where the segment joining the tip and the center of the base circle is not perpendicular to the base plane).

So what is the area of this, considering we know the radius, height and inclination?
 

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  • #2
well if you open up a right cone, you just get a circle with a notch out of it, and that is how you compute the area. so your examp[les aeems to yield an ellipse with a notch out of it, but the notch does not cut to the center nor any simple spot inside the elipse. so i do noit see immediately how to do it unless i have a parametrization of the ellipse.
 
  • #3
The surface area you desire will be the average of the surface areas for two right cones, one having a slant height equal to the shortest distance from the apex to the base (along the surface) and the other having a slant height equal to the greatest distance from the apex to the base (along the surface).
 
  • #4
Let your base be in the xy-plane with centre at the origin, and let the coordinates of your apex be:
[tex](a_{x},a_{y},a_{z})[/tex]
Then, the surface of the cone is parametrized by:
[tex]\vec{S}(z,\theta)=R(1-\frac{z}{a_{z}})((\cos\theta-\frac{a_{x}}{R})\vec{i}+(\sin\theta-\frac{a_{y}}{R})\vec{j})+(a_{x}\vec{i}+a_{y}\vec{j}+z\vec{k}), 0\leq\theta\leq2\pi,0\leq{z}\leq{a}_{z}[/tex]

Hence, the tangent vectors are:
[tex]\frac{\partial\vec{S}}{\partial\theta}=R(1-\frac{z}{a_{z}})(-\sin\theta\vec{i}+\cos\theta\vec{j})[/tex]
[tex]\frac{\partial\vec{S}}{\partial{z}}=-\frac{R}{a_{z}}((\cos\theta-\frac{a_{x}}{R})\vec{i}+(\sin\theta-\frac{a_{y}}{R})\vec{j})+\vec{k}[/tex]
The area is then given by:
[tex]A=\int_{0}^{2\pi}\int_{0}^{a_{z}}||\frac{\partial\vec{S}}{\partial\theta}\times\frac{\partial\vec{S}}{\partial{z}}||dzd\theta[/tex]
 

1. What is the formula for finding the lateral area of a cone?

The formula for finding the lateral area of a cone is L = πrs, where L is the lateral area, π is pi (approximately 3.14), r is the radius of the base, and s is the slant height of the cone.

2. How do you calculate the slant height of a cone?

The slant height of a cone can be calculated using the Pythagorean theorem: s = √(r2 + h2), where s is the slant height, r is the radius of the base, and h is the height of the cone.

3. Can the lateral area of a cone be negative?

No, the lateral area of a cone cannot be negative as it represents the surface area of the curved part of the cone, which cannot have a negative value.

4. How is the lateral area of a cone different from its total surface area?

The lateral area of a cone only includes the curved surface, while the total surface area includes both the curved surface and the area of the base. The formula for finding the total surface area of a cone is A = πr(r + s), where A is the total surface area, r is the radius of the base, and s is the slant height.

5. Can the lateral area of a cone be larger than its total surface area?

No, the lateral area of a cone cannot be larger than its total surface area as the lateral area is only a part of the total surface area.

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