Base Radius of Cone given Total Surface Area and Slant Height Calculator

Base Radius of Cone Formula:

\[ r_{Base} = \frac{1}{2} \times \left( \sqrt{h_{Slant}^2 + \frac{4 \times TSA}{\pi}} - h_{Slant} \right) \]

Base Radius of Cone:

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1. What is the Base Radius of Cone Formula?

The Base Radius of Cone formula calculates the radius of the circular base when given the slant height and total surface area of a cone. This geometric relationship is derived from the surface area formula of a cone.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_{Base} = \frac{1}{2} \times \left( \sqrt{h_{Slant}^2 + \frac{4 \times TSA}{\pi}} - h_{Slant} \right) \]

Where:

\( r_{Base} \) — Base Radius of Cone (m)
\( h_{Slant} \) — Slant Height of Cone (m)
\( TSA \) — Total Surface Area of Cone (m²)
\( \pi \) — Archimedes' constant (approximately 3.14159)

Explanation: The formula rearranges the total surface area equation to solve for the base radius, incorporating the square root function to handle the geometric relationship.

3. Importance of Base Radius Calculation

Details: Calculating the base radius is essential for various geometric and engineering applications, including volume calculations, structural design, and manufacturing processes involving conical shapes.

4. Using the Calculator

Tips: Enter slant height in meters, total surface area in square meters. Both values must be positive numbers greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between base radius and slant height?
A: The base radius, slant height, and height of the cone form a right triangle, related by the Pythagorean theorem: \( h_{Slant}^2 = r_{Base}^2 + h^2 \).

Q2: Can this formula be used for truncated cones?
A: No, this formula is specifically for complete right circular cones. Truncated cones have different surface area formulas.

Q3: What are typical units for these measurements?
A: While meters are used here, any consistent unit system can be applied (cm, mm, inches, etc.) as long as all measurements use the same units.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of cones, though real-world measurements may have practical limitations.

Q5: What if I only have the height instead of slant height?
A: You would need to use the Pythagorean theorem to find the slant height first: \( h_{Slant} = \sqrt{r_{Base}^2 + h^2} \).