A Sphere Has A Diameter { (d)$}$ Of 25 Mm. Find The Volume { (V)$}$ Of The Sphere In ML. (For A Sphere: ${V = \frac{4}{3} \pi R^3}$ And ${d = 2r}$)

Mar 13, 2025 by ADMIN 150 views

**A Sphere's Volume: Calculating the Volume of a Sphere with a Given Diameter**

Introduction

In mathematics, the volume of a sphere is a fundamental concept that has numerous applications in various fields, including physics, engineering, and computer science. The volume of a sphere is a measure of the amount of space inside the sphere. In this article, we will discuss how to calculate the volume of a sphere given its diameter. We will use the formula for the volume of a sphere, which is $V = \frac{4}{3} \pi r^3$ , where $r$ is the radius of the sphere.

Understanding the Formula

The formula for the volume of a sphere is $V = \frac{4}{3} \pi r^3$ . This formula is derived from the fact that the volume of a sphere is proportional to the cube of its radius. The constant of proportionality is $\frac{4}{3} \pi$ . To calculate the volume of a sphere, we need to know its radius. However, we are given the diameter of the sphere, which is $d = 25$ mm. We can use the fact that the diameter of a sphere is twice its radius to find the radius.

Finding the Radius

The diameter of a sphere is twice its radius, so we can write the equation $d = 2r$ . We are given that the diameter of the sphere is $d = 25$ mm, so we can substitute this value into the equation to get $25 = 2r$ . To find the radius, we can divide both sides of the equation by 2, which gives us $r = \frac{25}{2} = 12.5$ mm.

Calculating the Volume

Now that we have found the radius of the sphere, we can use the formula for the volume of a sphere to calculate the volume. We will substitute the value of the radius into the formula to get $V = \frac{4}{3} \pi (12.5)^3$ . To evaluate this expression, we can use a calculator or perform the calculations by hand.

Performing the Calculations

To calculate the volume of the sphere, we need to evaluate the expression $V = \frac{4}{3} \pi (12.5)^3$ . We can start by cubing the value of the radius, which gives us $(12.5)^3 = 1953.125$ . Then, we can multiply this value by $\frac{4}{3} \pi$ to get $V = \frac{4}{3} \pi (1953.125) = 8209.375 \pi$ . To find the numerical value of the volume, we can use the value of $\pi$ as approximately 3.14159.

Evaluating the Expression

To evaluate the expression $V = 8209.375 \pi$ , we can multiply the value of $\pi$ by 8209.375. This gives us $V = 8209.375 \times 3.14159 = 25851.419$ . Therefore, the volume of the sphere is approximately 25851.419 mm $^3$ .

Converting to Milliliters

To convert the volume of the sphere from cubic millimeters to milliliters, we can divide the volume by 1000. This gives us $\frac{25851.419}{1000} = 25.851419$ mL.

Conclusion

In this article, we have discussed how to calculate the volume of a sphere given its diameter. We have used the formula for the volume of a sphere, which is $V = \frac{4}{3} \pi r^3$ , and have found the radius of the sphere using the fact that the diameter of a sphere is twice its radius. We have then used the formula to calculate the volume of the sphere and have converted the result from cubic millimeters to milliliters. The volume of the sphere is approximately 25.851419 mL.

References

[1] "Sphere" by Math Open Reference. Retrieved from https://www.mathopenref.com/sphere.html
[2] "Volume of a Sphere" by Math Is Fun. Retrieved from https://www.mathsisfun.com/geometry/sphere-volume.html

Additional Resources

[1] "Sphere" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/Sphere.html
[2] "Volume of a Sphere" by Khan Academy. Retrieved from https://www.khanacademy.org/math/geometry/geometry-volume/volume-of-a-sphere

Q: What is the formula for the volume of a sphere?

A: The formula for the volume of a sphere is $V = \frac{4}{3} \pi r^3$ , where $r$ is the radius of the sphere.

Q: How do I find the radius of a sphere if I know its diameter?

A: To find the radius of a sphere if you know its diameter, you can use the fact that the diameter of a sphere is twice its radius. So, if the diameter is $d$ , the radius is $r = \frac{d}{2}$ .

Q: Can I use the formula for the volume of a sphere if I know the diameter instead of the radius?

A: Yes, you can use the formula for the volume of a sphere if you know the diameter instead of the radius. You can substitute the value of the diameter into the formula and then divide by 2 to get the radius.

Q: How do I calculate the volume of a sphere in cubic millimeters?

A: To calculate the volume of a sphere in cubic millimeters, you can use the formula $V = \frac{4}{3} \pi r^3$ and then multiply the result by 1000 to convert it to cubic millimeters.

Q: Can I use a calculator to calculate the volume of a sphere?

A: Yes, you can use a calculator to calculate the volume of a sphere. Simply enter the value of the radius and the formula $V = \frac{4}{3} \pi r^3$ into the calculator and press the "calculate" button.

Q: What is the volume of a sphere with a diameter of 25 mm?

A: To find the volume of a sphere with a diameter of 25 mm, you can use the formula $V = \frac{4}{3} \pi r^3$ and substitute the value of the radius, which is $r = \frac{25}{2} = 12.5$ mm. The result is $V = \frac{4}{3} \pi (12.5)^3 = 8209.375 \pi$ cubic millimeters.

Q: Can I convert the volume of a sphere from cubic millimeters to milliliters?

A: Yes, you can convert the volume of a sphere from cubic millimeters to milliliters by dividing the volume by 1000. So, if the volume is 8209.375 cubic millimeters, the volume in milliliters is $\frac{8209.375}{1000} = 8.209375$ milliliters.

Q: What are some real-world applications of the volume of a sphere?

A: The volume of a sphere has numerous real-world applications, including:

Calculating the volume of a ball or a sphere in a game or a simulation
Determining the volume of a container or a tank
Calculating the volume of a sphere in a scientific experiment or a study
Designing a sphere or a spherical object for a specific purpose

Q: Can I use the formula for the volume of a sphere to calculate the volume of a sphere with a non-integer radius?

A: Yes, you can use the formula for the volume of a sphere to calculate the volume of a sphere with a non-integer radius. Simply substitute the value of the radius into the formula and perform the calculations.

Q: What is the relationship between the volume of a sphere and its surface area?

A: The volume of a sphere is related to its surface area by the formula $V = \frac{4}{3} \pi r^3$ and $A = 4 \pi r^2$ . The surface area of a sphere is proportional to the square of its radius, while the volume of a sphere is proportional to the cube of its radius.

Q: Can I use the formula for the volume of a sphere to calculate the volume of a sphere with a negative radius?

A: No, you cannot use the formula for the volume of a sphere to calculate the volume of a sphere with a negative radius. The radius of a sphere must be a positive value, as it represents the distance from the center of the sphere to its surface.

Q: What is the significance of the value of pi (π) in the formula for the volume of a sphere?

A: The value of pi (π) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. In the formula for the volume of a sphere, pi (π) is used to calculate the volume of the sphere. The value of pi (π) is approximately 3.14159, but it can be approximated to any desired level of precision.