Volume of a Quarter Sphere

Before learning the formula for finding the volume of a quarter sphere, first, let us recall what is a sphere. A sphere is a perfectly symmetrical round-shaped three-dimensional object which resembles a globe, a ball, etc. The volume of a sphere is the amount of space that is inside the sphere. Thus, the volume of a quarter sphere is one-fourth of the volume of a full sphere.

Let us derive the formula for the volume of a quarter sphere and apply it to solve a few practice questions.

A quarter sphere is exactly one-fourth portion of a full sphere. i.e., if we divide a sphere into 4 equal parts, each portion is called a quarter sphere. Hence, the volume of a quarter sphere is one-fourth of the volume of a sphere.

Volume of a Quarter Sphere Using Radius

We know that the volume of a sphere of radius 'r' is V = (4/3)πr³. We have already seen that the volume of a quarter sphere is exactly one-fourth of the volume of a full of sphere. Thus,
The volume of a quarter sphere in terms of radius = 1/4 [(4/3)πr³] = (1/3)πr³.

Volume of a Quarter Sphere Using Diameter

Let us assume that the diameter of the sphere is d. Then we know that d = 2r (or) r = d/2. Substituting this in the above formula,

The volume of a quarter sphere in terms of diameter = (1/3)π(d/2)³ = (1/24)πd³.

Note: Here, π is a mathematical constant which is approximately equal to 22 / 7 (or) 3.14159...

Let us assume that r, d, A, and V be the radius, diameter, surface area, and volume of a sphere. Then here are the ways to find the volume of the quarter sphere depending on the available information.

• If r is given, straight away substitute it in the formula (1/3)πr³.
• If d is given then either find r using r = d/2 and use the formula (1/3)πr³ (or) directly substitute the value of d in the formula (1/24)πd³.
• If A is given, then use the surface area of a sphere formula, which is, A = πr², solve it for 'r' and then use the formula (1/3)πr³.
• If V is given then simply find V/4.

Now, that we have understood the formula and method to find the volume of a quarter sphere, let us have a look at a few solved examples for better understanding.

What Is a Quarter Sphere?

A quarter sphere is one-fourth portion of a sphere. i.e., when a sphere is divided equally into 4 parts, each part is a quarter sphere.

What Is the Volume of a Quarter Sphere Formula?

The volume of a quarter sphere is one-fourth of the volume of a sphere. Thus, the volume of a quarter sphere of radius 'r' is 1/4 [(4/3)πr³] = (1/3)πr³.

What Is the Formula Used To Find Volume of a Quarter Sphere?

The formula of finding the volume of a sphere of radius 'r' is obtained just by multiplying the volume of a full sphere by 1/4. i.e., the volume of a quarter sphere = 1/4 [(4/3)πr³] = (1/3)πr³.

What Is the Volume of a Quarter Sphere in Terms of Radius?

We know that the volume of a full sphere of radius 'r' is (4/3)πr³. So the volume of a quarter sphere is (1/3)πr³.

What Is the Volume of a Quarter Sphere in Terms of Diameter?

Consider a sphere of radius 'r' and diameter 'd'. Then the volume of the quarter sphere is 1/4 [(4/3)πr³] = (1/3)πr³. We know that r = d/2. Thus, the volume of a quarter sphere formula in terms of diameter is (1/3)π(d/2)³ = (1/24)πd³.

How To Find the Volume of a Quarter Sphere Given the Surface Area of the Sphere?

We know that the surface area of a sphere of radius 'r' is, A = 4πr². If A is given, then solve this equation for 'r' and then substitute it in the formula (1/3)πr³ to find the volume of the quarter sphere.

How To Find the Volume of a Quarter Sphere Given the Volume of the Sphere?

A quarter sphere is the one-fourth portion of a sphere. i.e., If the volume of a sphere is given, then we just divide it by 4 to find the volume of the quarter sphere.