What is a Sphere?
- A sphere is a perfectly symmetrical three-dimensional shape where every point on its surface is equidistant from its center.
- This distance from the center to any point on the surface is known as the radius (r) of the sphere. In simpler terms, a sphere looks like a ball, and its surface is uniformly curved with no edges or vertices.
Formula for Volume of Sphere:
- The formula to calculate the volume of a sphere is derived from integral calculus and is fundamental in geometry.
- It is always expressed in cubic units.
The Volume(V) of the sphere is given by,
V=\frac{4}{3}\pi r^3
Where,
r is the radius of the sphere.
Solving the Equation for Radius r.
If we know the Volume(V) of a sphere, we can solve for the radius.
V=\frac{4}{3}\pi r^3
Divide both sides by \frac{4}{3}\pi
\frac{V}{\frac{4}{3}\pi }=r^3\\
\frac{3V}{4\pi }=r^3\\
r^3=\frac{3V}{4\pi }
Take a square root on both sides.
r=\left(\frac{3V}{4\pi }\right)^{\frac{1}{3}}
Thus, given the Volume V, you can determine the radius r of the sphere using this formula.
Properties of a Sphere
- The sphere is a three-dimensional shape with no edges or vertices.
- Every point on the surface of a sphere is equidistant from the center.
- The volume of a sphere is proportional to the cube of its radius.
- A sphere has the largest volume among all closed surfaces enclosing the same surface area.
Applications
- Physics: Calculating volumes in physical phenomena and thermodynamics.
- Astronomy: Determining the volume of celestial bodies like planets and stars.
- Engineering: Designing spherical structures and optimizing space utilization.
Solved Examples
Example 1: Calculate the volume of a sphere with a radius of 5 cm.
Solution:
V=\frac{4}{3}\pi r^3\\
Substitute r=5
V=\frac{4}{3}\pi \cdot 5^3\\
V=\frac{4}{3}\pi \cdot 125\\
V=\frac{500\pi }{3}\\
The volume of a sphere is \frac{500\pi }{3} cubic centimeters.
Example 2: A sphere has a volume of 904.32 cubic meters. Find its radius.
Solution:
r=\left(\frac{3V}{4\pi }\right)^{\frac{1}{3}}\\
Substitute V=904.32
r=\left(\frac{3\cdot 904.32}{4\pi }\right)^{\frac{1}{3}}\\
r=\left(216\right)^{\frac{1}{3}}\\
r=6\\
The radius of the sphere is 6 m.
FAQs on the Volume of the sphere
1) What does the volume of a sphere represent?
Answer: The volume of a sphere quantifies the three-dimensional space it occupies.
2) What are the units for the volume of a sphere?
Answer: The volume of a sphere is expressed in cubic units. For example, if the radius is in cm, the volume will be in cubic centimeters (cm3); if the radius is in meters, the volume will be in cubic meters (m3).
3) How does the volume of a sphere change if the radius is doubled
Answer: If the radius of a sphere is doubled, the volume increases by a factor of 2^3=8. This is because the volume formula involves the cube of the radius:
V=\frac{4}{3}\pi \left(2r\right)^3\\
V=\frac{4}{3}\pi \cdot 8r^3\\
V=8\left(\frac{4}{3}\pi r^3\right)\\
