Volume of a Sphere | MathHelp.com

Summary

This video explains how to calculate the volume of a sphere using the formula V = 4/3 * π * r³. By substituting the radius (3 inches) into the formula, the calculation simplifies to 36π cubic inches. The video further demonstrates how to approximate the volume by replacing π with 3.14, resulting in a volume of approximately 113.04 cubic inches. Viewers are reminded of the importance of understanding the formula and how to apply it to determine the volume of any sphere.

Takeaways

• 😀 The formula for the volume of a sphere is 4/3 * π * r³.
• 😀 The radius of the sphere in this example is 3 inches.
• 😀 To calculate the volume, substitute the radius (3 inches) into the formula.
• 😀 3 inches cubed (3³) equals 27 cubic inches.
• 😀 After substitution, the formula becomes 4/3 * π * 27 cubic inches.
• 😀 Simplify the expression by canceling out 27 and 3 to get 9.
• 😀 This results in 4 * π * 9 cubic inches, which simplifies further to 36 * π cubic inches.
• 😀 To get a numerical approximation, substitute π with 3.14.
• 😀 The volume is approximately 36 * 3.14 = 113.04 cubic inches.
• 😀 The final volume of the sphere is 113.04 cubic inches when approximating π.
• 😀 This process demonstrates how to apply the sphere volume formula and approximate using 3.14 for π.

Q & A

What is the formula for the volume of a sphere?

The formula for the volume of a sphere is V = (4/3) * π * r³, where r is the radius of the sphere.

How do we calculate the volume of a sphere if the radius is 3 inches?

To calculate the volume, substitute the radius (3 inches) into the formula: V = (4/3) * π * (3)³. After cubing 3 inches (which is 27), the volume becomes (4/3) * π * 27, simplifying to 36π cubic inches.

What does cubing a number mean, and how is it used in the calculation?

Cubing a number means multiplying it by itself twice. In this case, 3 inches cubed is 3 * 3 * 3 = 27 inches³. This value is used to find the volume of the sphere.

How does the simplification of 27 and 3 lead to the final volume formula?

When calculating (4/3) * π * 27, the 27 and 3 can cancel each other out, leaving 9. This simplifies the formula to 36π cubic inches.

Why is Pi approximated as 3.14 in this calculation?

Pi (π) is an irrational number, meaning it cannot be expressed exactly as a decimal. For practical purposes, it is commonly approximated as 3.14 to simplify calculations.

How is the volume of the sphere expressed after using the approximation for Pi?

After substituting Pi with 3.14, the volume formula becomes 36 * 3.14, which equals 113.04 cubic inches.

What is the significance of the units used in the volume calculation?

The units of the volume are cubic inches (in³) because the volume is a measure of three-dimensional space, and each dimension is measured in inches.

Can the volume formula for a sphere be used with any radius?

Yes, the volume formula for a sphere can be used with any radius, as long as the radius is known. The formula adjusts based on the value of the radius.

What happens if the radius of the sphere is doubled? How does this affect the volume?

If the radius is doubled, the volume of the sphere will increase by a factor of eight, because the volume depends on the cube of the radius (r³). So, doubling the radius results in a volume that is 2³ = 8 times larger.

Is there a difference between the exact volume formula and the approximate volume found using Pi = 3.14?

Yes, there is a difference. The exact volume would use the true value of Pi, but using Pi = 3.14 provides an approximation. The exact volume would be slightly larger than 113.04 cubic inches.