Height of Cone given Total Surface Area and Base Area Calculator

Formula Used:

\[ h = \sqrt{\left(\frac{TSA}{\sqrt{\pi \cdot A_{Base}}} - \sqrt{\frac{A_{Base}}{\pi}}\right)^2 - \frac{A_{Base}}{\pi}} \]

Height of Cone (h):

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1. What is the Height of Cone Formula?

The height of a cone can be calculated using the total surface area and base area through the derived mathematical formula. This formula provides a way to determine the vertical height when the surface measurements are known.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ h = \sqrt{\left(\frac{TSA}{\sqrt{\pi \cdot A_{Base}}} - \sqrt{\frac{A_{Base}}{\pi}}\right)^2 - \frac{A_{Base}}{\pi}} \]

Where:

\( h \) — Height of Cone (m)
\( TSA \) — Total Surface Area of Cone (m²)
\( A_{Base} \) — Base Area of Cone (m²)
\( \pi \) — Archimedes' constant (approximately 3.14159)

Explanation: The formula derives the height by relating the total surface area to the base area through geometric relationships in a right circular cone.

3. Importance of Height Calculation

Details: Calculating the height of a cone is essential in various engineering, architectural, and mathematical applications where dimensional analysis is required for construction, volume calculations, or geometric modeling.

4. Using the Calculator

Tips: Enter the total surface area and base area in square meters. Both values must be positive numbers greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What units should I use for the inputs?
A: The calculator uses square meters (m²) for both surface area and base area inputs. Ensure consistent units for accurate results.

Q2: Does this formula work for all types of cones?
A: This formula is specifically designed for right circular cones where the apex is directly above the center of the circular base.

Q3: What if I get a negative value under the square root?
A: A negative value indicates invalid input combinations where the given surface area and base area cannot form a valid cone geometry.

Q4: How accurate is this calculation?
A: The calculation uses double-precision floating-point arithmetic and provides results accurate to six decimal places.

Q5: Can I use this for truncated cones?
A: No, this formula is specifically for complete cones. Different formulas are required for truncated cones (frustums).