Diameter of a Sphere Formula Using Volume
The volume of a sphere formula can be found in terms of the diameter. The diameter of a sphere is the longest line that is inside the sphere and that passes through the center of the sphere. A sphere is a threedimensional shape that is perfectly symmetrical and round in shape. Some examples of spheres are a ball, a globe, etc. The volume of a sphere is the amount of space that is inside it (or) the capacity of the sphere that it can hold. In this article, we will derive the diameter of a sphere formula using the volume.
How to Find Volume of a Sphere with Diameter?
We know that the volume, V of a sphere of radius 'r' is calculated using the formula V = (4/3) π r^{3}. This is the formula for the volume of a sphere in terms of radius. We know that the diameter of a sphere is twice the radius. i.e., the diameter, d = 2r. From this, we get r = d/2. Substituting this in the volume formula,
V = (4/3) π (d/2)^{3} = (4/3) π (d^{3}/8) = (πd^{3})/6.
Thus the volume of a sphere in terms of diameter (d) is, V = (πd^{3})/6.
What Is the Diameter of a Sphere Formula Using Volume?
In the previous section, we have already learned the formula of finding the volume (V) of a sphere using its diameter (d). We can simply solve this formula for the diameter (d) to find the diameter of a sphere formula using its volume. The volume of a sphere using diameter is,
V = (πd^{3})/6
⇒ 6V = πd^{3} (Multiplied both sides by 6)
⇒ 6V/π = d^{3} (Divided both sides by π)
⇒ (6V/π)^{(1/3)} = d (raised the power on both sides by 1/3)
Thus, the formula for the diameter of a sphere using its volume is, d = (6V/π)^{(1/3)}.
Note: Here π is a constant which is approximately equal to 22/7 (or) 3.14159...
How to Calculate the Diameter of a Sphere Using Volume?
Let us consider a sphere of radius r, diameter d, and volume V. We can calculate the diameter of the sphere using its volume in 3 ways:
 Substitute the value of V in the volume of a sphere formula in terms of radius, V = (4/3) π r^{3}, solve it for r. Then use d = 2r.
 Substitute the value of V in the volume of a sphere formula in terms of diameter, V = (πd^{3})/6, solve it for d.
 Directly substitute the value of V in the diameter of a sphere formula using volume, d = (6V/π)^{(1/3)}.
Solved Examples on Diameter of a Sphere Formula Using Volume

Example 1: The volume of a spherical ball is 36π cubic units. Find its diameter.
Solution:
The volume of the given spherical ball is, V = 36π cubic units.
Method 1
Substituting this in the diameter formula using volume,
d = (6V/π)^{(1/3)} = (6(36π) /π)^{(1/3)} = (216)^{(1/3)} = 6 units.
Method 2
Using the formula of volume of a sphere,
V = (4/3) π r^{3}
⇒ 36π = (4/3) π r^{3}
⇒ 36π × (3/4) × (1/π) = r^{3}
⇒ 27 = r^{3}
⇒ r = 3So the radius of the sphere is r = 3. From this, its diameter = 2(3) = 6 units.
Answer: The diameter of the given spherical ball = 6 units.

Example 2: The volume of a sphere is 28π cm^{3}. Find the diameter of the sphere using volume.
Solution
The volume of the given sphere is, V = 28π cubic units.
Substituting this in the diameter of sphere formula using volume,
d = (6V/π)^{(1/3)} = (6(28π) /π)^{(1/3)} = (168)^{(1/3)} = (8)^{(1/3) }× (21)^{(1/3) } = 2 \(\sqrt[3]{21}\).
Answer: The diameter of the given sphere = 2 \(\sqrt[3]{21}\) cm.
Practice Questions on Diameter of a Sphere Formula Using Volume
FAQs on Diameter of a Sphere Formula Using Volume
How To Calculate the Volume of a Sphere With Diameter?
The volume of a sphere (V) in terms of its radius (r) is V = (4/3) π r^{3}. If d is its diameter, we have d = 2r. From this, we get r = (d/2). Substituting this in the volume formula, the volume of a sphere in terms of diameter is V = (πd^{3})/6.
How Do You Find the Diameter of a Sphere From the Volume?
There are multiple methods to find the diameter (d) of a sphere from the volume (V). One of the methods is to substitute V directly into the formula, d = (6V/π)^{(1/3)}. Alternatively, we can substitute V into the volume of a sphere formula, V = (4/3) π r^{3}, solve for r, and double the radius (r) to find the diameter.
What Is the Diameter of a Sphere Formula Using Volume?
The diameter of a sphere formula with volume is, d = (6V/π)^(1/3), where d is the diameter of the sphere and V is its volume. Alternatively, we can substitute the value of V, in the volume formula, V = (4/3) π r^{3}, solve it for r, and make it double to find the diameter.
What Is the Relation Between the Diameter and Volume of a Sphere?
The relationship between the diameter of a sphere and its volume can be studied from the formula, V = (πd^{3})/6, where, d is the diameter and V s the volume of the sphere.
What Is the Formula to Calculate the Radius and Diameter of Sphere Using Volume?
The radius and diameter of a sphere can be calculated, given the volume. The formulas that can be used to find the radius and diameter are given below,
 Formula for radius, r of sphere using Volume, V = (4/3) π r^{3}.
 Formula for diameter, d of sphere using Volume, V = (πd^{3})/6.
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