Volume of Cone given Total Surface Area and Base Area Calculator

Formula Used:

\[ V = \frac{A_{Base} \times \sqrt{\left(\frac{TSA}{\sqrt{\pi \times A_{Base}}} - \sqrt{\frac{A_{Base}}{\pi}}\right)^2 - \frac{A_{Base}}{\pi}}}{3} \]

Volume of Cone:

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1. What is the Volume of Cone Formula?

The volume of a cone given total surface area and base area is calculated using a specialized formula that relates these geometric properties. This formula provides an accurate measurement of the three-dimensional space enclosed by the cone's surface.

2. How Does the Calculator Work?

The calculator uses the following formula:

\[ V = \frac{A_{Base} \times \sqrt{\left(\frac{TSA}{\sqrt{\pi \times A_{Base}}} - \sqrt{\frac{A_{Base}}{\pi}}\right)^2 - \frac{A_{Base}}{\pi}}}{3} \]

Where:

\( V \) — Volume of Cone (m³)
\( A_{Base} \) — Base Area of Cone (m²)
\( TSA \) — Total Surface Area of Cone (m²)
\( \pi \) — Mathematical constant pi (approximately 3.14159)

Explanation: The formula derives the volume by relating the base area and total surface area through geometric relationships involving the cone's dimensions.

3. Importance of Volume Calculation

Details: Calculating the volume of a cone is essential in various fields including engineering, architecture, manufacturing, and mathematics education. It helps determine capacity, material requirements, and spatial relationships in conical structures.

4. Using the Calculator

Tips: Enter base area and total surface area in square meters. Both values must be positive numbers. The calculator will compute the volume in cubic meters.

5. Frequently Asked Questions (FAQ)

Q1: What units should I use for the inputs?
A: The calculator expects base area and total surface area in square meters, and returns volume in cubic meters.

Q2: Why might I get an error message?
A: An error occurs when the input values result in a negative expression under the square root, which indicates geometrically impossible dimensions.

Q3: Can this formula be used for truncated cones?
A: No, this formula is specifically for complete cones. Different formulas apply to truncated cones (frustums).

Q4: How accurate is the calculation?
A: The calculation uses double-precision arithmetic and provides results accurate to 6 decimal places.

Q5: What if I only know the radius and slant height?
A: You would need to use different formulas that directly use radius and slant height to calculate volume.