Top Radius of Solid of Revolution Calculator

Top Radius of Solid of Revolution Formula:

\[ r_{Top} = \sqrt{\frac{TSA - LSA}{\pi}} - r_{Bottom} \]

Total Surface Area of Solid of Revolution (TSA):

m²

Lateral Surface Area of Solid of Revolution (LSA):

m²

Bottom Radius of Solid of Revolution (r_Bottom):

Top Radius of Solid of Revolution (r_Top):

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1. What is Top Radius of Solid of Revolution?

The Top Radius of Solid of Revolution is the horizontal distance from the top end point of the revolving curve to the axis of rotation of the Solid of Revolution. It's a key parameter in calculating the dimensions of 3D objects created by rotating a 2D curve around an axis.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_{Top} = \sqrt{\frac{TSA - LSA}{\pi}} - r_{Bottom} \]

Where:

\( r_{Top} \) — Top Radius of Solid of Revolution (m)
\( TSA \) — Total Surface Area of Solid of Revolution (m²)
\( LSA \) — Lateral Surface Area of Solid of Revolution (m²)
\( r_{Bottom} \) — Bottom Radius of Solid of Revolution (m)
\( \pi \) — Archimedes' constant (approximately 3.14159)

Explanation: The formula calculates the top radius by first finding the area of the top circular surface (TSA - LSA), converting it to radius using the circle area formula, and then subtracting the bottom radius.

3. Importance of Top Radius Calculation

Details: Calculating the top radius is essential in engineering design, architectural planning, and manufacturing processes where solids of revolution are involved. It helps determine the complete dimensions of conical, cylindrical, or other rotational symmetric objects.

4. Using the Calculator

Tips: Enter total surface area and lateral surface area in square meters, and bottom radius in meters. All values must be positive, and total surface area must be greater than lateral surface area.

5. Frequently Asked Questions (FAQ)

Q1: What types of solids does this formula apply to?
A: This formula applies to solids of revolution where the top and bottom surfaces are circular, such as truncated cones, cylinders, and similar rotational symmetric shapes.

Q2: Why is π used in the formula?
A: π is used because we're dealing with circular surfaces. The area of a circle is πr², so we need π to convert between area and radius.

Q3: What if my solid doesn't have a circular top?
A: This formula specifically applies to solids with circular top surfaces. For other shapes, different formulas would be needed.

Q4: Can this calculator handle different units?
A: The calculator uses meters for length and square meters for area. Convert your measurements to these units before calculation.

Q5: What if I get a negative result?
A: A negative result typically indicates invalid input values, such as lateral surface area being larger than total surface area, or the bottom radius being too large for the given surface areas.