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

Formula Used:

\[ h = \sqrt{l^2 - \left( \sqrt{\frac{A_{\text{top}}}{\pi}} - \sqrt{\frac{A_{\text{base}}}{\pi}} \right)^2} \]

Height (h):

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1. What is the 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 a crucial measurement in geometry and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ h = \sqrt{l^2 - \left( \sqrt{\frac{A_{\text{top}}}{\pi}} - \sqrt{\frac{A_{\text{base}}}{\pi}} \right)^2} \]

Where:

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

Explanation: The formula derives from the Pythagorean theorem, using the slant height and the difference in radii of the two bases to find the perpendicular height.

3. Importance of Height Calculation

Details: Calculating the height of a frustum is essential in various fields including architecture, engineering, and manufacturing where conical sections are used. It helps in determining volume, surface area, and structural properties.

4. Using the Calculator

Tips: Enter the slant height and areas of both circular bases. All values must be positive numbers. The calculator will compute the perpendicular height of the frustum.

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 formula be used for any frustum?
A: Yes, this formula applies to any right circular conical frustum where the cutting plane is parallel to the base.

Q3: What units should I use?
A: Use consistent units for all measurements (e.g., all in meters or all in inches). The height will be in the same linear units.

Q4: What if the slant height is less than the difference in radii?
A: This would result in a negative value under the square root, indicating an impossible geometric configuration. Please verify your inputs.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact, assuming precise input values and using the exact value of π in computations.