Height of Cone given Lateral Surface Area Calculator

Formula Used:

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

Height of Cone (h):

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1. What is the Height of Cone given Lateral Surface Area Formula?

The formula calculates the height of a cone when its lateral surface area and base radius are known. It is derived from the relationship between the lateral surface area, base radius, and slant height of a cone.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( h \) — Height of Cone (m)
\( LSA \) — Lateral Surface Area of Cone (m²)
\( r_{Base} \) — Base Radius of Cone (m)
\( \pi \) — Archimedes' constant (approximately 3.1416)

Explanation: The formula calculates the height by first finding the slant height from the lateral surface area, then using the Pythagorean theorem to find the vertical height.

3. Importance of Height Calculation

Details: Calculating the height of a cone is essential in geometry, engineering, and architecture for determining volume, surface area, and other geometric properties of conical structures.

4. Using the Calculator

Tips: Enter lateral surface area in square meters, base radius in meters. Both values must be positive numbers greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is the lateral surface area of a cone?
A: The lateral surface area is the area of the curved surface of the cone, excluding the base area.

Q2: Can this formula be used for truncated cones?
A: No, this formula is specifically for right circular cones. Truncated cones require a different formula.

Q3: What units should I use for the inputs?
A: Use consistent units (e.g., meters for length, square meters for area). The calculator will output height in the same length unit as the radius input.

Q4: What if I get a negative value under the square root?
A: A negative value indicates invalid input where the lateral surface area is too small for the given base radius, which is geometrically impossible.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for perfect right circular cones, assuming precise input values.