# Deriving Formula for Radius of Conical Section at Height h

• zorro
In summary, to find the formula for the radius at a height 'h' on a conical section with a total height 'l' and radii 'a' and 'b' (where a>b), one can draw a cross section and look for similar triangles. Alternatively, one can think of the conical section as a trapezoid and use the equation r = a - h(a-b)/l to find the radius at height 'h'.
zorro

## Homework Statement

Suppose there is a Conical section (of a right circular cone) of total height 'l' and radii 'a' and 'b' (a>b). How do we derive the formula for the radius at a height 'h' (h<l) ?

## The Attempt at a Solution

Last edited:
Draw a cross section through the middle and look for similar triangles whose properties you can use to solve your problem. (I think you are describing a frustrated cone, not a pure cone?)

I am talking about this (see attached file).

#### Attachments

• cone.jpg
6.5 KB · Views: 389
pongo38 said:
Draw a cross section through the middle and look for similar triangles whose properties you can use to solve your problem. (I think you are describing a frustrated cone, not a pure cone?)
Good advice, but the term is "frustum of a cone."

Mark44 said:
Good advice, but the term is "frustum of a cone."

Haha (or in other words, a truncated cone)
How was his advice good? We can't use similar triangle property to find the radius.

Extend the vertical line at the center and the outer sloped line up until they meet, then you'll have similar triangles.

Forget the 3 dimensional aspect; it would be easier to think of a trapezoid with one side of length [itex]l[/tex], perpendicular to sides a & b. Call the remaining side c.

a & b still represent the upper and lower radii of your truncated cone, and r represents the radius of that object at height h above side a.

#### Attachments

• Trapezoid4.png
1.3 KB · Views: 399
Mark44 said:
Extend the vertical line at the center and the outer sloped line up until they meet, then you'll have similar triangles.

I find the algebra easier if you construct the additional 2 line segments shown in the attached drawing.

#### Attachments

• Trapezoid5.png
1.3 KB · Views: 428
Thanks a lot zgozvrm. I got my answer

Glad to help.

What did you come up with?

r = a - h(a-b)/l

Very nice!

## 1. What is the formula for the radius of a conical section at a given height?

The formula for the radius of a conical section at a given height is r = h * tan(θ), where r is the radius, h is the height, and θ is the angle of the cone.

## 2. How is the formula for the radius of a conical section derived?

The formula for the radius of a conical section is derived using basic trigonometry principles, specifically the tangent function. The cone can be divided into two right triangles, and the tangent of the angle between the height and base of the cone is equal to the ratio of the opposite side (radius) to the adjacent side (height).

## 3. Can the formula for the radius of a conical section be used for any type of cone?

Yes, the formula for the radius of a conical section can be used for any type of cone, as long as the angle of the cone remains constant. This means that the shape of the cone may vary, but the angle between the height and base must remain the same.

## 4. How accurate is the formula for the radius of a conical section?

The formula for the radius of a conical section is accurate as long as the angle of the cone remains constant. However, there may be slight variations due to factors such as the curvature of the cone and the precision of measurements.

## 5. Can the formula for the radius of a conical section be used for other shapes?

No, the formula for the radius of a conical section is specific to cones and cannot be used for other shapes. Other shapes may have their own formulas for determining their radius at a given height.

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