Spherical Sector Formula

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_bh. 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 = 2πR²h/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 = 2πR²h/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 = 2πR²h/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 = 2πR²h/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)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.

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)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.

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)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.

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 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.

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)πR²

If θ is in degrees then volume of a spherical wedge = (θ/360°)(4/3)πR²

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)πR². 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)πR²
• Step 4: Represent the final answer in cubic units.

Solved Examples on Section of a Sphere

What Is the Volume of Section of a Sphere Formula?

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)πR², where, angle θ (in radians) formed by the wedge and its radius R.

How to Calculate Volume of a Spherical Segment?

is base radius, R${}_2$ is radius of top circle, and h is height of spherical segment.

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