# Finding the volume of the cone using cylindrical polar coordinates?

• sarubobo28
In summary, the cone surface is defined by the equation ρ+z=1 and the limits for ρ and z are 0 to (1-z) and 0 to 1, respectively. This is because the defining equation for the cone relates ρ and z, and using 0 to 1 for both would define a cylinder instead.
sarubobo28
The cone centre is the z-axis and has base ρ=1 and height z=1,
I'm looking at the lecture notes and it says the limit φ=0 to 2pi, z=0 to 1,
ρ=0 to (1-z).
Could someone tell me where the (1-z) comes from please?
Why is it not 0 to 1?

On the cone surface, ρ and z are related through ρ+z=1. This is the defining equation for that surface. 0 to 1 would define a cylinder, which is equally wide in top and bottom.

clamtrox said:
On the cone surface, ρ and z are related through ρ+z=1. This is the defining equation for that surface. 0 to 1 would define a cylinder, which is equally wide in top and bottom.

I see, thank you I get it now :)

## 1. What is the formula for finding the volume of a cone using cylindrical polar coordinates?

The formula for finding the volume of a cone using cylindrical polar coordinates is V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone.

## 2. How do you convert cylindrical polar coordinates to Cartesian coordinates?

To convert cylindrical polar coordinates (r, θ, z) to Cartesian coordinates (x, y, z), you can use the following equations:
x = rcosθ
y = rsinθ
z = z

## 3. Can you find the volume of a cone using only polar coordinates?

Yes, the volume of a cone can be found using only polar coordinates. The formula for finding the volume using polar coordinates is V = (1/3)πr^2h, where r is the distance from the origin to the point on the base and h is the height of the cone.

## 4. How do you find the height of a cone using polar coordinates?

The height of a cone can be found using the formula h = z/r, where z is the vertical distance from the base to the apex of the cone and r is the distance from the origin to the point on the base.

## 5. Can you use cylindrical polar coordinates to find the volume of a cone with a slanted base?

Yes, cylindrical polar coordinates can be used to find the volume of a cone with a slanted base. The formula for finding the volume would be V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone measured along the slant.

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