Height of Cone given Lateral Surface Area and Base Circumference Calculator

Formula Used:

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

Height of Cone (h):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Height of Cone Formula?

The Height of Cone formula calculates the vertical distance from the apex to the base center of a cone using its lateral surface area and base circumference. This geometric relationship helps determine the cone's dimensions when certain parameters are known.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( h \) — Height of Cone (m)
\( LSA \) — Lateral Surface Area of Cone (m²)
\( C_{Base} \) — Base Circumference of Cone (m)
\( \pi \) — Archimedes' constant (≈3.14159)

Explanation: The formula derives from the relationship between lateral surface area, base circumference, and the slant height of the cone, using the Pythagorean theorem to find the vertical height.

3. Importance of Height Calculation

Details: Calculating the height of a cone is essential in various applications including architecture, manufacturing, packaging design, and geometric modeling where precise dimensional measurements are required.

4. Using the Calculator

Tips: Enter lateral surface area in square meters, base circumference in 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 inputs?
A: Use consistent units (meters for length, square meters for area). The calculator will output height in the same length unit as the base circumference input.

Q2: Can this formula be used for truncated cones?
A: No, this formula is specifically for right circular cones. Different formulas apply for truncated cones (frustums).

Q3: What if I have the radius instead of circumference?
A: You can convert radius to circumference using \( C = 2\pi r \), or use the alternative height formula: \( h = \sqrt{\left(\frac{LSA}{\pi r}\right)^2 - r^2} \)

Q4: How accurate is this calculation?
A: The calculation is mathematically exact for perfect right circular cones. Real-world accuracy depends on the precision of your input measurements.

Q5: What are typical height values for cones?
A: Cone heights vary widely depending on application - from millimeters in small containers to meters in architectural structures. There are no fixed "normal" values.