Outer Radius of Hollow Sphere given Surface Area Calculator

Formula Used:

\[ r_{Outer} = \sqrt{\frac{SA}{4\pi} - r_{Inner}^2} \]

Outer Radius of Hollow Sphere (r_Outer):

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1. What is the Outer Radius of Hollow Sphere Formula?

The formula calculates the outer radius of a hollow sphere when the surface area and inner radius are known. It's derived from the surface area formula of a hollow sphere, which accounts for both the inner and outer surfaces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_{Outer} = \sqrt{\frac{SA}{4\pi} - r_{Inner}^2} \]

Where:

\( r_{Outer} \) — Outer Radius of Hollow Sphere (m)
\( SA \) — Surface Area of Hollow Sphere (m²)
\( r_{Inner} \) — Inner Radius of Hollow Sphere (m)
\( \pi \) — Archimedes' constant (approximately 3.14159)

Explanation: The formula rearranges the surface area equation of a hollow sphere to solve for the outer radius, using the known surface area and inner radius values.

3. Importance of Outer Radius Calculation

Details: Calculating the outer radius is essential in engineering and manufacturing applications involving hollow spherical objects, such as pressure vessels, storage tanks, and architectural structures. It helps determine material requirements and structural integrity.

4. Using the Calculator

Tips: Enter surface area in square meters and inner radius in meters. Both values must be positive numbers, and the inner radius squared must be less than SA/(4π) for a valid result.

5. Frequently Asked Questions (FAQ)

Q1: What is a hollow sphere?
A: A hollow sphere is a three-dimensional object with a spherical outer surface and a spherical inner cavity, creating a uniform thickness throughout.

Q2: Why is the formula structured this way?
A: The formula is derived from the surface area formula SA = 4π(r_Outer² + r_Inner²) and rearranged to solve for the outer radius.

Q3: What units should I use?
A: Use consistent units (preferably meters for radius and square meters for area) to ensure accurate results.

Q4: What if I get a negative value under the square root?
A: A negative value indicates invalid input where the inner radius squared exceeds SA/(4π), which is mathematically impossible for a hollow sphere.

Q5: Can this calculator handle different units?
A: The calculator assumes consistent units. Convert all measurements to the same unit system before calculation.