Height of Truncated Cone given Curved Surface Area Calculator

Formula Used:

\[ Height\ of\ Truncated\ Cone = \sqrt{\left(\frac{Curved\ Surface\ Area\ of\ Truncated\ Cone}{\pi \times (Base\ Radius\ of\ Truncated\ Cone + Top\ Radius\ of\ Truncated\ Cone)}\right)^2 - (Base\ Radius\ of\ Truncated\ Cone - Top\ Radius\ of\ Truncated\ Cone)^2} \]

Curved Surface Area of Truncated Cone:

m²

Base Radius of Truncated Cone:

Top Radius of Truncated Cone:

Height of Truncated Cone:

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1. What is the Height of Truncated Cone Formula?

The height of a truncated cone (frustum) can be calculated using its curved surface area and the radii of its top and bottom bases. This formula is derived from the geometry of the frustum and the Pythagorean theorem.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: The formula calculates the slant height component from the curved surface area, then uses the Pythagorean theorem to find the vertical height.

3. Importance of Height Calculation

Details: Calculating the height of a truncated cone is essential in various engineering, architectural, and manufacturing applications where precise dimensions are required for construction, volume calculations, or material estimation.

4. Using the Calculator

Tips: Enter all measurements in consistent units (meters recommended). Ensure the curved surface area is greater than zero, and both radii are positive values. The top radius should be smaller than the base radius for a proper frustum.

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 circular bases of different sizes.

Q2: Can this formula be used for any truncated cone?
A: Yes, as long as you have the curved surface area and both radii, this formula will calculate the height accurately.

Q3: What if I have the total surface area instead of curved surface area?
A: You would need to subtract the areas of both circular bases from the total surface area to get the curved surface area before using this calculator.

Q4: Are there any limitations to this formula?
A: The formula assumes a perfect right circular truncated cone and may not be accurate for irregular shapes or cones with non-parallel bases.

Q5: What units should I use?
A: You can use any consistent units (cm, m, inches, etc.), but the calculator assumes meters by default. Make sure all inputs use the same unit system.