Top Radius of Truncated Cone given Slant Height Calculator

Formula Used:

\[ r_{Top} = r_{Base} - \sqrt{h_{Slant}^2 - h^2} \]

Base Radius of Truncated Cone (rBase):

Slant Height of Truncated Cone (hSlant):

Height of Truncated Cone (h):

Top Radius of Truncated Cone (rTop):

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1. What is the Top Radius of Truncated Cone?

The Top Radius of a Truncated Cone is the radius of the smaller circular top surface of a frustum (truncated cone). It is an important geometric parameter used in various engineering and architectural applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_{Top} = r_{Base} - \sqrt{h_{Slant}^2 - h^2} \]

Where:

\( r_{Top} \) — Top Radius of Truncated Cone (m)
\( r_{Base} \) — Base Radius of Truncated Cone (m)
\( h_{Slant} \) — Slant Height of Truncated Cone (m)
\( h \) — Height of Truncated Cone (m)

Explanation: This formula calculates the top radius using the Pythagorean theorem applied to the cross-section of the truncated cone.

3. Importance of Top Radius Calculation

Details: Calculating the top radius is essential for determining the volume, surface area, and other geometric properties of truncated cones used in construction, manufacturing, and design applications.

4. Using the Calculator

Tips: Enter base radius, slant height, and height in meters. All values must be positive, and slant height must be greater than the height for valid results.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated cone?
A: A truncated cone (frustum) is a cone with the tip cut off by a plane parallel to the base, resulting in two parallel circular surfaces of different sizes.

Q2: When is this calculation useful?
A: This calculation is useful in engineering, architecture, and manufacturing where truncated cone shapes are common, such as in storage tanks, funnels, and architectural elements.

Q3: What units should be used?
A: The calculator uses meters (m) for all measurements, but the formula works with any consistent unit system.

Q4: What if the slant height is less than the height?
A: The calculation requires that slant height be greater than height, as this follows from the Pythagorean theorem. If slant height ≤ height, the result would be mathematically invalid.

Q5: Can this formula be used for other calculations?
A: Yes, this formula is part of a system of equations that can be rearranged to solve for other parameters of a truncated cone given different known values.