How To Find Radius Of Sphere With Volume

Savager Indie vor

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23-11-2024

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Knowing the volume of a sphere and wanting to find its radius is a common problem in geometry and various scientific applications. This article will guide you through the process, explaining the formula and providing example calculations. We'll cover both the mathematical derivation and practical applications.

Understanding the Formula

The volume (V) of a sphere is given by the formula:

V = (4/3)πr³

Where:

• V = Volume of the sphere
• π (pi) ≈ 3.14159
• r = Radius of the sphere

To find the radius (r) when you know the volume (V), we need to rearrange this formula. This involves a few algebraic steps.

Rearranging the Formula to Solve for Radius

1. Multiply both sides by 3: 3V = 4πr³

2. Divide both sides by 4π: (3V)/(4π) = r³

3. Take the cube root of both sides: ∛((3V)/(4π)) = r

Therefore, the formula to calculate the radius (r) of a sphere given its volume (V) is:

r = ∛((3V)/(4π))

Step-by-Step Calculation Examples

Let's work through a couple of examples to solidify your understanding.

Example 1: Finding the radius of a sphere with a volume of 100 cubic centimeters.

1. Substitute the known value into the formula: r = ∛((3 * 100 cm³)/(4π))

2. Calculate the expression inside the cube root: (300 cm³)/(4π) ≈ 23.87 cm³

3. Take the cube root: ∛(23.87 cm³) ≈ 2.88 cm

Therefore, the radius of the sphere is approximately 2.88 centimeters.

Example 2: A sphere has a volume of 36π cubic meters. Find its radius.

1. Substitute the known value into the formula: r = ∛((3 * 36π m³)/(4π))

2. Notice that π cancels out: r = ∛((108)/(4)) = ∛(27)

3. Take the cube root: ∛(27) = 3

Therefore, the radius of this sphere is 3 meters. This example highlights how simplifying the equation first can make calculations easier.

Practical Applications

The ability to calculate the radius of a sphere from its volume has numerous applications across various fields:

• Engineering: Designing spherical tanks, containers, or components.
• Physics: Calculating the properties of spherical objects, such as atoms or planets.
• Astronomy: Determining the size of celestial bodies.
• Medicine: Analyzing the size of cells or tumors.

Troubleshooting and Common Mistakes

• Units: Ensure consistent units throughout the calculation. If the volume is in cubic meters, the radius will be in meters.
• Calculator Use: Use a calculator that can handle cube roots and π accurately.
• Rounding: Round your final answer to an appropriate number of significant figures based on the precision of the given volume.

By understanding the formula and following the steps outlined above, you can confidently calculate the radius of any sphere given its volume. Remember to double-check your work and pay close attention to units for accurate results.