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## Spherical Sector

A spherical sector is a solid generated by revolving a sector of a circle about an axis which passes through the center of the circle but which contains no point inside the sector. If the axis of revolution is one of the radial sides, the sector thus formed is spherical cone; otherwise, it is open spherical sector.

**Properties of Spherical Sector**

- Spherical sector is bounded by a zone and one or two conical surfaces.
- The spherical sector having only one conical surface is called a
*spherical cone*, otherwise it is called*open spherical sector*. - The
*base*of spherical sector is its zone.

**Formulas for Spherical SectorTotal surface area, A**

The total surface area of a spherical sector is equal to the area of the zone plus the sum of the lateral areas of the bounding cones.

Total surface area = Zone + Lateral area of bounding cones

A=Azone+A1+A2

A=2πRh+πaR+πbR

A=πR(2h+a+b)

Note that for spherical cone, b = 0 and the equation will reduce to

A=πR(2h+a)

**Volume, V**

The volume of spherical sector, either open spherical sector or spherical cone, is equal to one-third of the product of the area of the zone and the radius of the sphere. This is similar to the volume of a cone which is V_{cone} = 1/3 A_{b}h. In spherical sector, replace A_{b} with A_{zone} and h with R.

## Spherical Sector Formula

A spherical sector is a solid formed by rotating a sector of a circle along an axis that passes through the circle’s centre but has no points inside the sector. If the revolution axis is one of the radial sides, the resulting sector is a spherical cone; otherwise, it is an open spherical sector. A zone and one or two conical surfaces surround a spherical sector. A spherical sector is that part of the sphere that has a vertex at the centre and a conical boundary.

**Spherical Sector Formulas**

A spherical sector has two major formulae that are connected to its area and volume.

**Total surface area formula (A)**

The sum of the area of the zone and the lateral areas of the bounding cones is equal to the total surface area of a spherical sector.

A = Area of zone + Lateral area of bounding cones

A = 2πRh + πaR + πbR

A = πR (2h + a + b)Now, for a spherical cone, b = 0. So, the area in that case is,

A = πR (2h + a)

**Volume of a spherical sector formula**

The volume of a spherical sector, whether open or conical, equals one-third of the product of the area of the zone and the radius of the sphere.

V = (1/3) × Area of zone × radius

V = (1/3) × 2πRh × R

V = 2πR^{2}h/3

**Sample Problems**

**Problem 1. Find the total surface area of a sector of a sphere if its radius is 6 cm, the radius of the bounding cone is 8 cm and the height is 10 cm.**

**Solution:**

We have, R = 6, h = 10 and a = 8.

Using the formula we have,

A = πR (2h + a)

= (22/7) (6) (20 + 8)

= (22/7) (6) (28)

= 528 sq. cm

**Problem 2. Find the radius of the spherical sector if its total surface area is 2640 sq. cm, the radius of the bounding cone is 9 cm and the height is 7 cm.**

**Solution:**

We have, V = 2640, h = 7 and a = 9.

Using the formula we have,

A = πR (2h + a)

=> 2640 = (22/7) (R) (14 + 9)

=> 2640 = (22/7) (R) (23)

=> 18480 = 506R

=> R = 18480/506

=> R = 36.52 cm

**Problem 3. Find the volume of the spherical sector if its radius is 21 cm and height is 18 cm.**

**Solution:**

We have, R = 21 and h = 18.

Using the formula we have,

V = 2πR

^{2}h/3= 2 (22/7) (21) (21) (18) (1/3)

= 2 (22) (3) (21) (6)

= 16632 cu. cm

**Problem 4. Find the height of the spherical sector if its volume is 660 cu. cm and radius is 12 cm.**

**Solution:**

We have, V = 660 and R = 12.

Using the formula we have,

V = 2πR

^{2}h/3=> 660 = 2 (22/7) (12) (12) (h) (1/3)

=> 660 = 2112h/7

=> 301.71h = 660

=> h = 2.18 cm

**Problem 5. Find the volume of the spherical sector if its total surface area is 132 sq. cm, height is 7 cm and radius of the bounding cone is 4 cm.**

**Solution:**

We have, A = 132, h = 7 and a = 4.

Using the formula we have,

A = πR (2h + a)

=> 132 = (22/7) (R) (18)

=> 396R = 924

=> R = 2.33

So, the volume is,

V = 2πR

^{2}h/3= 2 (22/7) (2.33) (2.33) (7) (1/3)

= 238.87/3

= 79.62 cu. cm

## Sphere

A **sphere** is a solid bounded by a surface all points of which are equally distant a point within, called *center*.

A **radius** is the distance between the center and the surface.

A **diameter** of a sphere is a straight line drawn through the center, having its extremities in the surface.

### Formulas

**Surface area:**

## Spherical Sector

A **spherical sector** is a portion of a sphere defined by a conical boundary with apex at the center of the sphere.

The curved surface area of the spherical sector (on the surface of the sphere, excluding the cone surface) is:

A=2πRh

Total surface area (including the cone surface):

A=πR(2h+r)A=πR(2h+r)

**Volume:**

#### Spherical cap (Spherical segment of one base)

A **spherical cap** is a portion of a sphere cut off by a plane.

The curved surface area of the spherical cap:

## Spherical segment

A **spherical segment** is a portion of the sphere included between two parallel planes.

The curved surface area of the spherical zone – which excludes the top and bottom bases:

Curved surface area =2πRh=2πRh

The surface area – which includes the top and bottom bases:

## 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.

## What Is Volume of Section of a Sphere?

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

## Volume of a Spherical Cap Formula

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)πh ^{2}(3R – h)**

By using Pythagoras theorem, (R – h)^{2} + a^{2} = R^{2}

Therefore, volume can be rewritten as, Volume of a spherical cap in terms of h and a = (1/6)πh(3a^{2} + h^{2})

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.

### How to Find the Volume of a Spherical Cap?

As we learned in the previous section, the volume of the spherical cap is (1/3)πh^{2}(3R – h) or (1/6)πh(3a^{2} + h^{2}). 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}+ a^{2}= R^{2}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)πh^{2}(3R – h) or V = (1/6)πh(3a^{2}+ h^{2}).**Step 5:**Represent the final answer in cubic units.

## Volume of a Spherical Sector (Spherical Cone)

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)πR ^{2}h**

### 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) πR^{2}h. 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}+ a^{2}= R^{2}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)πR^{2}h.**Step 5:**Represent the final answer in cubic units.

## Volume of a Spherical Segment (Spherical Frustum)

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 **R _{1}** of the base circle of the segment, and radius

**R**of the top circle of the segment, then its volume can be given by:

_{2}**Volume of a spherical segment = (1/6)πh(3R _{1}^{2} + 3R_{2}^{2} + h^{2})**

### 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(3R_{1}^{2} + 3R_{2}^{2} + h^{2}). 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 R_{1}and identify the radius of the top circle and name this radius as R_{2}**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(3R_{1}^{2}+ 3R_{2}^{2}+ h^{2})**Step 4:**Represent the final answer in cubic units.

## Volume of a Spherical Wedge

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)πR ^{2}**

If θ is in degrees then volume of a spherical wedge = (θ/360°)(4/3)πR^{2}

### 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)πR^{2}. 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)πR^{2}**Step 4:**Represent the final answer in cubic units.

## Solved Examples on Section of a Sphere

**Example 1:** Find the volume of a spherical cap having base radius = 7 units and height = 21 units using a section of a sphere formula. (Use π = 22/7)

**Solution**

Base radius of the spherical cap (a) = 7 units

Height of the spherical cap (h) = 21 units

Volume of the spherical cap = (1/6)πh(3a^{2} + h^{2}) = (1/6) × (22/7) × 21 × (3 × 7^{2} + 21^{2}) = 11 × (3 × 49 + 441) = 6468 units^{3}

**Answer: **Volume of the spherical cap = 6468 units^{3}

**Example 2:** Find the volume of the spherical cone if the height of its spherical cap = 7 units and the radius of the original sphere = 9 units. (Use π = 22/7)

**Solution**

Height of the spherical cap = 7 units

Radius of the original sphere = 9 units

Using section of a sphere formula,

Volume of the spherical cone = (2/3)πR^{2}h = (2/3) × (22/7) × 9^{2} × 7 = 1188 units^{3}

**Answer:** Volume of the spherical cone = 1188 units^{3}

## FAQs on the Volume of Section of a Sphere

### 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)πh
^{2}(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)πR
^{2}h, where, R is radius of sphere, h is height. - Volume of a spherical segment = (1/6)πh(3R
_{1}^{2}+ 3R_{2}^{2}+ h^{2}), where, R

is base radius, R

- is radius of top circle, and h is height of spherical segment.
- Volume of a spherical wedge = (θ/2π)(4/3)πR
^{2}, 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)πh^{2}(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)πh^{2}(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(3R_{1}^{2} + 3R_{2}^{2} + h^{2}), where, R

is base radius, R 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)πR^{2}, where, angle θ (in radians) formed by the wedge and its radius R.

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