Height of Cone given Total Surface Area and Base Circumference Calculator

Formula Used:

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

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 circumference with the formula:

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

This formula allows you to find the vertical height of a cone when you know its total surface area and the circumference of its base.

2. How Does the Calculator Work?

The calculator uses the derived formula:

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

Where:

\( h \) — Height of Cone (m)
\( TSA \) — Total Surface Area of Cone (m²)
\( C_{Base} \) — Base Circumference of Cone (m)
\( \pi \) — Mathematical constant pi (approximately 3.14159)

Explanation: The formula is derived from the relationships between the cone's surface area, base circumference, slant height, and vertical height.

3. Importance of Cone Height Calculation

Details: Calculating the height of a cone is essential in various geometric applications, engineering designs, architectural planning, and manufacturing processes where conical shapes are involved.

4. Using the Calculator

Tips: Enter the total surface area in square meters and base circumference in meters. Both values must be positive numbers. The calculator will compute the height of the cone.

5. Frequently Asked Questions (FAQ)

Q1: What units should I use for the inputs?
A: Use consistent units (preferably meters for length and square meters for area) to get the height in meters.

Q2: What if I get an imaginary number result?
A: This indicates that the input values are not physically possible for a real cone. Check your measurements.

Q3: Can I use this formula for truncated cones?
A: No, this formula is specifically for right circular cones, not truncated cones (frustums).

Q4: How accurate is this calculation?
A: The calculation is mathematically exact, but the accuracy depends on the precision of your input measurements.

Q5: What's the relationship between slant height and vertical height?
A: Slant height (l) relates to vertical height (h) and base radius (r) through the Pythagorean theorem: \( l = \sqrt{h^2 + r^2} \).