Height of Truncated Cone given Total Surface Area Calculator

Formula Used:

\[ h = \sqrt{\left(\frac{TSA - \pi(r_{base}^2 + r_{top}^2)}{\pi(r_{base} + r_{top})}\right)^2 - (r_{base} - r_{top})^2} \]

Total Surface Area of Truncated Cone (TSA):

m²

Base Radius of Truncated Cone (r_base):

Top Radius of Truncated Cone (r_top):

Height of Truncated Cone (h):

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1. What is the Height of Truncated Cone Calculation?

The height of a truncated cone (frustum) given its total surface area is calculated using a specific geometric formula that relates the height to the total surface area and the radii of the base and top circular surfaces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ h = \sqrt{\left(\frac{TSA - \pi(r_{base}^2 + r_{top}^2)}{\pi(r_{base} + r_{top})}\right)^2 - (r_{base} - r_{top})^2} \]

Where:

\( h \) — Height of Truncated Cone (m)
\( TSA \) — Total Surface Area of Truncated Cone (m²)
\( r_{base} \) — Base Radius of Truncated Cone (m)
\( r_{top} \) — Top Radius of Truncated Cone (m)
\( \pi \) — Pi constant (approximately 3.14159)

Explanation: The formula derives the height by considering the relationship between the total surface area, the lateral surface area, and the geometric properties of the truncated cone.

3. Importance of Height Calculation

Details: Calculating the height of a truncated cone is essential in various engineering, architectural, and manufacturing applications where precise dimensional measurements are required for conical frustum shapes.

4. Using the Calculator

Tips: Enter total surface area in square meters, base radius in meters, and top radius in meters. All values must be positive numbers. Ensure the top radius is smaller than the base radius for a valid truncated cone.

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 construction, manufacturing, and design where conical frustum shapes are used, such as in storage tanks, piping systems, and architectural elements.

Q3: What are the limitations of this formula?
A: The formula assumes perfect geometric shapes and may not account for material thickness or manufacturing tolerances in real-world applications.

Q4: Can this calculator handle different units?
A: The calculator uses meters for length units. For other units, convert all measurements to meters before calculation.

Q5: What if I get an undefined result?
A: An undefined result typically occurs when the denominator becomes zero, which happens when the sum of base and top radii is zero, indicating invalid input values.