Technology
Calculating the Volume of a Sphere from Its Circumference
Introduction to Spherical Geometry
Spherical geometry is a non-Euclidean geometry that studies the properties of figures on the surface of a sphere. A sphere is a three-dimensional object where every point on its surface is equidistant from its center. The circumference of a sphere is the total distance around its largest horizontal cross-section, which is a circle.
Understanding the Formulas
Two key formulas are necessary to solve this problem: the formula for the circumference of a sphere and the formula for its volume.
Circumference of a Sphere
The circumference of a sphere can be calculated using the formula: C 2πr
VOLUME OF A SPHERE
The volume of a sphere can be found using the formula: V frac{4}{3}πr^3
Example Problem
Given the circumference of a sphere is 18π inches, we will find its volume in terms of π. Let's break this down step by step.
Step 1: Determine the Radius
Using the circumference formula, we can find the radius r of the sphere.
Given: C 2πr 18π 2πr
Divide both sides of the equation by 2π to isolate r: r frac{18π}{2π} 9 text{ inches}
Step 2: Calculate the Volume
Now that we have the radius, we can find the volume.
Using the volume formula: V frac{4}{3}πr^3
Substitute the Value of Radius
Substitute r 9 text{ inches} V frac{4}{3}π(9)^3
Simplify the Expression
Calculate the value of 9^3 9^3 729
Substitute this value back into the formula: V frac{4}{3}π(729) frac{2916}{3}π 972π
Final Volume
Conclusion
Therefore, the volume of the sphere is 972π cubic inches.
Additional Insights
This method can be applied to any sphere where the circumference is given. Understanding these formulas and their application is crucial for solving problems in spherical geometry. Remember, the key is to accurately apply the formulas and simplify your calculations step by step.