Height of Cone given Slant Height and Base Circumference Calculator

Formula Used:

\[ Height of Cone = \sqrt{Slant Height of Cone^2 - \left(\frac{Base Circumference of Cone}{2\pi}\right)^2} \] \[ h = \sqrt{h_{Slant}^2 - \left(\frac{C_{Base}}{2\pi}\right)^2} \]

Height of Cone:

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1. What is the Height of Cone Formula?

The Height of Cone formula calculates the vertical height of a right circular cone when given its slant height and base circumference. This formula is derived from the Pythagorean theorem applied to the cross-section of the cone.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( h \) — Height of Cone (m)
\( h_{Slant} \) — Slant Height of Cone (m)
\( C_{Base} \) — Base Circumference of Cone (m)
\( \pi \) — Archimedes' constant (approximately 3.14159)

Explanation: The formula calculates the height by first converting the base circumference to radius, then applying the Pythagorean theorem to the right triangle formed by the height, radius, and slant height.

3. Importance of Height Calculation

Details: Calculating the height of a cone is essential in various applications including geometry problems, architectural design, manufacturing processes, and volume calculations for conical objects.

4. Using the Calculator

Tips: Enter slant height and base circumference in meters. Both values must be positive numbers. The calculator will compute the vertical height of the cone.

5. Frequently Asked Questions (FAQ)

Q1: What units should I use for input values?
A: The calculator uses meters for both input and output. Ensure consistent units for accurate results.

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

Q3: What if the calculated height is imaginary?
A: An imaginary result indicates that the given measurements cannot form a valid cone - the slant height must be greater than the radius derived from the base circumference.

Q4: Can I use diameter instead of circumference?
A: Yes, but you would need to convert diameter to circumference first (Circumference = π × Diameter).

Q5: How accurate is the calculation?
A: The calculation is mathematically exact for the given inputs, using the precise value of π for computations.