Calculating the Volume of a Sphere Given Its Greatest Circle Circumference
Understanding how to calculate the volume of a sphere using its greatest circle circumference is a fundamental concept in geometry. In this article, we will explain the process step-by-step and provide practical examples to ensure clarity.
Introduction to Great Circle Circumference
A great circle on the surface of a sphere is a circle that divides the sphere into two equal hemispheres. It is essentially the largest possible circle that can be drawn on a sphere, and it passes through the sphere's center. The circumference of a great circle is directly related to the sphere's radius, playing a crucial role in determining various geometric properties of the sphere.
Given Information and Solution
We are given that the circumference of the greatest circle of a sphere is 180 cm. Our goal is to find the volume of the sphere. Let's break down the solution step-by-step:
Circumference of a Circle Formula:
The circumference ( C ) of a circle is given by the formula:
[ C 2 pi r ]where ( r ) is the radius of the circle. In this case, the circumference of the greatest circle is 180 cm. So, we can write the equation as:
[ 180 2 pi r ]To find the radius ( r ), we solve for ( r ):
[ r frac{180}{2 pi} frac{90}{pi} ]Note that the value of ( pi approx 3.14159 ), so:
[ r approx frac{90}{3.14159} approx 28.6479 text{ cm} ]Volume of a Sphere Formula:
The volume ( V ) of a sphere is given by the formula:
[ V frac{4}{3} pi r^3 ]Substitute the value of the radius ( r ) into this formula:
[ V frac{4}{3} pi (28.6479)^3 ]Now, we calculate ( (28.6479)^3 ):
[ (28.6479)^3 approx 235351.432 ]Therefore:
[ V approx frac{4}{3} pi times 235351.432 ][ V approx frac{4}{3} times 3.14159 times 235351.432 ][ V approx 984841.905 ]Thus, the volume of the sphere is approximately 98484.190 cm3.
Conclusion
By following the steps mentioned above, we have calculated the volume of a sphere given its greatest circle circumference. This method is straightforward and can be applied to similar problems involving geometric shapes and measurements. Understanding such concepts is not only useful in academic settings but also in real-world applications such as engineering and architecture.