Height of Frustum of Cone given Slant Height and Top Area Calculator

Formula Used:

\[ h = \sqrt{l^2 - \left(\sqrt{\frac{A_{Top}}{\pi}} - r\right)^2} \]

Height (h):

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1. What is Height of Frustum of Cone?

The height of a frustum of a cone is the perpendicular distance between its two parallel circular bases. It is an important geometric measurement used in various applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ h = \sqrt{l^2 - \left(\sqrt{\frac{A_{Top}}{\pi}} - r\right)^2} \]

Where:

\( h \) — Height of the frustum
\( l \) — Slant height of the frustum
\( A_{Top} \) — Area of the top circular base
\( r \) — Radius of the bottom circular base
\( \pi \) — Mathematical constant (approximately 3.14159)

Explanation: This formula calculates the height by first determining the top radius from the given top area, then using the Pythagorean theorem with the slant height and the difference between top and bottom radii.

3. Importance of Height Calculation

Details: Calculating the height of a frustum is essential in geometry, engineering, architecture, and various practical applications where conical frustum shapes are used, such as in containers, structural elements, and manufacturing components.

4. Using the Calculator

Tips: Enter the slant height, top area, and base radius. All values must be positive numbers. The top area should be greater than zero, and the slant height should be greater than the absolute difference between top and bottom radii.

5. Frequently Asked Questions (FAQ)

Q1: What is a frustum of a cone?
A: A frustum is the portion of a cone that remains after cutting off the top by a plane parallel to the base.

Q2: Can this calculator handle different units?
A: Yes, as long as consistent units are used for all measurements (e.g., all in centimeters or all in inches).

Q3: What if the calculated height is imaginary?
A: An imaginary result indicates that the given measurements cannot form a valid frustum (e.g., slant height is too short for the given radii).

Q4: How accurate is the calculation?
A: The calculation is mathematically exact based on the input values, using the precise value of π.

Q5: Can this formula be used for truncated pyramids?
A: No, this formula is specific to conical frustums. Pyramids have different geometric properties.