Volume of a Section of a Sphere

In this section, we will discuss the volume of a section of a sphere along with solved examples. Let us start with the pre-required knowledge to understand the topic, volume of a section of a sphere. The volume of a three-dimensional object is defined as the space occupied by the object in a three-dimensional space.

Volume of section of sphere is defined as the total space occupied by a section of the sphere. A section of a sphere is a portion of a sphere. In other words, it is the shape obtained when the sphere is cut in a specific way. The section of a sphere can have various possible shapes depending on how it is cut. Spherical sector, spherical cap, spherical segment, and spherical wedge are well-known examples of a section of a sphere. Let us see the formulas to calculate volume of these different types of sections of sphere,

• Volume of spherical cap
• Volume of spherical sector
• Volume of spherical segment
• Volume of spherical wedge

A spherical cap is a portion of a sphere obtained when the sphere is cut by a plane. For a sphere, if the following are given: height h of the spherical cap, radius a of the base circle of the cap, and radius R of the sphere (from which the cap was removed), then its volume can be given by:
Volume of a spherical cap in terms of h and R = (1/3)πh²(3R - h)

By using Pythagoras theorem, (R - h)² + a² = R²

Therefore, volume can be rewritten as, Volume of a spherical cap in terms of h and a = (1/6)πh(3a² + h²)

For a spherical cap having a height equal to the radius, h = R, then it is a hemisphere.

Note: The range of values for the height is 0 ≤ h ≤ 2R and range of values for the radius of the cap is 0 ≤ a ≤ R.

As we learned in the previous section, the volume of the spherical cap is (1/3)πh²(3R - h) or (1/6)πh(3a² + h²). Thus, we follow the steps shown below to find the volume of the spherical cap.

• Step 1: Identify the radius of the sphere from which the spherical cap was taken from and name this radius as R.
• Step 2: Identify the radius of the spherical cap and name it as a or the height of the spherical and name it as h.
• Step 3: You can use the relation (R - h)² + a² = R² if any two of the variables are given and the third is unknown.
• Step 4: Find the volume of the spherical cap using the formula, V = (1/3)πh²(3R - h) or V = (1/6)πh(3a² + h²).
• Step 5: Represent the final answer in cubic units.

A spherical sector is a portion of a sphere that consists of a spherical cap and a cone with an apex at the center of the sphere and the base of the spherical cap. The volume of a spherical sector can be said as the sum of the volume of the spherical cap and the volume of the cone. For a spherical sector, if the following are given: height h of the spherical cap, radius a of the base circle of the cap, and radius R of the sphere (from which the cap was removed), then its volume can be given by:

Volume of a spherical cone in terms of h and R = (2/3)πR²h

How to Find the Volume of a Spherical Sector (Spherical Cone)?

As we learned in the previous section, the volume of the spherical sector is (2/3) πR²h. Thus, we follow the steps shown below to find the volume of the spherical sector.

• Step 1: Identify the radius of the sphere from which the spherical sector was taken and name this radius as R.
• Step 2: Identify the radius of the spherical cap and name it as a or the height of the spherical cap and name it as h.
• Step 3: You can use the relation (R - h)² + a² = R² if any two of the variables are given and the third is unknown.
• Step 4: Find the volume of the spherical sector using the formula V = (2/3)πR²h.
• Step 5: Represent the final answer in cubic units.

A spherical sector is a portion of a sphere that is obtained when a plane cuts the sphere at the top and bottom such that both the cuts are parallel to each other. For a spherical segment, if the following are given: height h of the spherical segment, radius R₁ of the base circle of the segment, and radius R₂ of the top circle of the segment, then its volume can be given by:

Volume of a spherical segment = (1/6)πh(3R₁² + 3R₂² + h²)

How To Find the Volume of a Spherical Segment (Spherical Frustum)?

As we learned in the previous section, the volume of the spherical segment is (1/6)πh(3R₁² + 3R₂² + h²). Thus, we follow the steps shown below to find the volume of the spherical segment.

• Step 1: Identify the radius of the base circle and name this radius as R₁ and identify the radius of the top circle and name this radius as R₂
• Step 2: Identify the height of the spherical segment and name it as h.
• Step 3: Find the volume of the spherical sector using the formula V = (1/6)πh(3R₁² + 3R₂² + h²)
• Step 4: Represent the final answer in cubic units.

A solid formed by revolving a semicircle about its diameter with less than 360 degrees. For a spherical wedge, if the following are given: angle θ (in radians) formed by the wedge and its radius R, then its volume can be given by:

Volume of a spherical wedge = (θ/2π)(4/3)πR³

If θ is in degrees then volume of a spherical wedge = (θ/360°)(4/3)πR³

How To Find the Volume of a Spherical Wedge?

As we learned in the previous section, the volume of the spherical wedge is (θ/2π)(4/3)πR³. Thus, we follow the steps shown below to find the volume of the spherical wedge.

• Step 1: Identify the radius of the spherical wedge and name it as R.
• Step 2: Identify the angle of the spherical wedge and name it as θ.
• Step 3: Find the volume of the spherical wedge using the formula, V = (θ/2π)(4/3)πR³
• Step 4: Represent the final answer in cubic units.

What Is Meant By Volume of Section of a Sphere?

The total space occupied by a section of the sphere is called the volume of a section of a sphere. A section of a sphere is a portion of a sphere. The volume of a section of a sphere is expressed in square units.

What Is the Volume of Section of a Sphere Formula?

The formulas to calculate the volume of different types of section of a sphere,

• Volume of a spherical cap = (1/3)πh²(3R - h), where, height h of the spherical cap, and radius R of the sphere from which cap was cut.
• Volume of a spherical sector = (2/3)πR²h, where, R is radius of sphere, h is height.
• Volume of a spherical segment = (1/6)πh(3R₁² + 3R₂² + h²), where, R\(_1\) is base radius, R\(_2\) is radius of top circle, and h is height of spherical segment.
• Volume of a spherical wedge = (θ/2π)(4/3)πR³, where, angle θ (in radians) formed by the wedge and its radius R.

How Do You Find the Volume of a Section of a Sphere?

We can calculate the volume of a section of a sphere using the formula, V = (1/3)πh²(3R - h), where, height h of the spherical section, and radius R of the sphere from which the section was cut.

What Is Volume of a Spherical Cap?

The volume of a spherical cap is given by the formula, Volume of a spherical cap = (1/3)πh²(3R - h), where, height h of the spherical cap, and radius R of the sphere from which cap was cut.

How to Calculate Volume of a Spherical Segment?

The volume of a spherical segment is given by the formula, Volume of a spherical segment = (1/6)πh(3R₁² + 3R₂² + h²), where, R\(_1\) is base radius, R\(_2\) is radius of top circle, and h is height of spherical segment.

What Is the Volume of Spherical Wedge?

The volume of a spherical wedge is given by the formula, Volume of a spherical wedge = (θ/2π)(4/3)πR³, where, angle θ (in radians) formed by the wedge and its radius R.