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.
| 1. | What Is Volume of Section of a Sphere? |
| 2. | Volume of a Spherical Cap Formula |
| 3. | Volume of a Spherical Sector (Spherical Cone) |
| 4. | Volume of a Spherical Segment (Spherical Frustum) |
| 5. | Volume of a Spherical Wedge |
| 6. | FAQs on Section of Sphere |
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.
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)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)πR2, 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)πR2, where, angle θ (in radians) formed by the wedge and its radius R.
FAQs
How do you find the volume of a section of a sphere? ›
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 example of volume of sphere? ›The formula for the volume of a sphere is V = 4/3 π r³, where V = volume and r = radius. The radius of a sphere is half its diameter. So, to calculate the surface area of a sphere given the diameter of the sphere, you can first calculate the radius, then the volume.
What is the section of a sphere? ›In geometry, a spherical sector, also known as a spherical cone, is a portion of a sphere or of a ball defined by a conical boundary with apex at the center of the sphere. It can be described as the union of a spherical cap and the cone formed by the center of the sphere and the base of the cap.
What is the section of a sphere by a plane? ›In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i.e., bounded by a single plane.
What are 2 examples of volume? ›For example, if a cup can hold 100 ml of water up to the brim, its volume is said to be 100 ml. Volume can also be defined as the amount of space occupied by a 3-dimensional object. The volume of a solid like a cube or a cuboid is measured by counting the number of unit cubes it contains.
What are 3 examples of a sphere? ›Marbles, balls, oranges, yarn, and bubbles are a few common examples of sphere shapes in real life.
What are three examples of volume? ›A volume is simply defined as the amount of space occupied by any three-dimensional solid. These solids can be a cube, a cuboid, a cone, a cylinder or a sphere.
What are two ways to find the volume of a sphere? ›The volume V of a sphere is four-thirds times pi times the radius cubed. The volume of a hemisphere is one-half the volume of the related sphere. Note : The volume of a sphere is 2/3 of the volume of a cylinder with same radius, and height equal to the diameter.
What is the formula to solve volume? ›The answer to a volume is shown in cubic units. The formula for volume is: Volume = length x width x height.
What is the area and volume of a sphere? ›| Sphere Formulas | |
|---|---|
| Diameter of a Sphere | D = 2 r |
| Surface Area of a Sphere | A = 4 π r2 |
| Volume of a Sphere | V = (4 ⁄ 3) π r3 |
What happens to the volume of the sphere? ›
Answer and Explanation: The volume is increased by 8 times when the diameter of the sphere is doubled. Formula : 43πr3 4 3 π r 3 . Here r refers to the radius.
What is sphere value? ›SPH, or sphere, refers to the amount of correction needed for nearsightedness or farsightedness. Optometrists measure SPH values in increments of 0.25 diopters, and when people talk about their prescription degree, they're usually talking about their sphere value.
What is sphere definition and formula? ›A sphere is defined by three axes, x-axis, y-axis and z-axis. The region occupied by a circle is simply an area. The formula of the area is πr2. A sphere has a surface area covered by its outer surface, which is equal to 4πr2. It does not have any volume.
Is a section of sphere a circle? ›All cross sections of a sphere are circles. (All circles are similar to one another.) If two planes are equidistant from the center of a sphere, and intersecting the sphere, the intersected circles are congruent.
What is sphere explained? ›A sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the centre of the sphere, and r is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.
What is every cross-section of a sphere? ›The intersection of a plane figure with a sphere is a circle. All cross-sections of a sphere are circles.
What is a section of a plane called? ›The main sections of an airplane include the fuselage, wings, cockpit, engine, propeller, tail assembly, and landing gear. Understanding the basic functions of how these parts interact is the first step to understanding the principles of aerodynamics.
What is a simple definition of volume? ›Volume is defined as the space occupied within the boundaries of an object in three-dimensional space. It is also known as the capacity of the object.
What is volume definition and formula? ›Volume is a measure of three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The definition of length (cubed) is interrelated with volume.
What are the 3 formulas for volume? ›| Table 3. Volume Formulas | ||
|---|---|---|
| Shape | Formula | Variables |
| Cube | V=s3 | s is the length of the side. |
| Right Rectangular Prism | V=LWH | L is the length, W is the width and H is the height. |
| Prism or Cylinder | V=Ah | A is the area of the base, h is the height. |
What are the 4 main spheres give an example of each? ›
These four subsystems are called "spheres." Specifically, they are the "lithosphere" (land), "hydrosphere" (water), "biosphere" (living things), and "atmosphere" (air). Each of these four spheres can be further divided into sub-spheres.
What are the 5 main spheres? ›The five systems of Earth (geosphere, biosphere, cryosphere, hydrosphere, and atmosphere) interact to produce the environments we are familiar with.
What is the sphere equation in 3D? ›Answer: The equation of a sphere in standard form is x2 + y2 + z2 = r2. Let us see how is it derived. Explanation: Let A (a, b, c) be a fixed point in the space, r be a positive real number and P (x, y, z ) be a moving point such that AP = r is a constant.
How do you measure volume example? ›You can measure out the length x width x depth of your living room to find its volume. The average living room is 15 feet by 10 feet, with a height of nine feet. This comes to a volume of 1,350 cubic feet.
What is an example of volume in a sentence? ›She fiddled with the volume on the stereo. a high volume of sales Huge volumes of park visitors come through every weekend. an increase in traffic volume The box has a volume of three cubic meters.
What is the equation of sphere example? ›The general equation of a sphere is: (x - a)² + (y - b)² + (z - c)² = r², where (a, b, c) represents the center of the sphere, r represents the radius, and x, y, and z are the coordinates of the points on the surface of the sphere.
What is the formula in finding the surface area of a sphere answer? ›And the formula for the surface area of a sphere of radius R is 4*Pi*R2. And, you can check that the latter is the derivative of the former with respect to R.
How do you find the radius of a sphere when you know the volume? ›- Multiply the volume by three.
- Divide the result by four times pi.
- Find the cube root of the result from Step 2.
- The result is your sphere's radius!
It's obtained by simply dividing the sum of all values in a data set by the number of values. The calculation can be done from raw data or for data aggregated in a frequency table.
How do you find the volume and surface area of a sphere of radius A? ›The formula for the volume of a sphere is V=43πr3, and the formula for the surface area of a sphere is A=4πr2.
Who found a way to calculate the volume of a sphere? ›
The way Archimedes found his formulas is both amazingly clever and shows him to be a mathematician of the first rank, far ahead of others of his time, doing mathematics within touching distance of integral calculus 1800 years before it was invented.
What is volume of a circle formula? ›Thus, the volume can be written as the product of the area of the circle and its thickness dy. Also, the radius of the circular disc “r” can be expressed in terms of the vertical dimension (y) using the Pythagoras theorem. V = 4 3 π R 3 c u b i c u n i t s .
What is surface area of sphere definition? ›The surface area of a sphere is defined as the region covered by its outer surface in three-dimensional space. A Sphere is a three-dimensional solid having a round shape, just like a circle. The formula of total surface area of a sphere in terms of pi (π) is given by: Surface area = 4 π r2 square units.
What is the formula for area and volume? ›Whereas the basic formula for the area of a rectangular shape is length × width, the basic formula for volume is length × width × height. How you refer to the different dimensions does not change the calculation: you may, for example, use 'depth' instead of 'height'.
How does volume of a sphere relate to surface area? ›The volume of a sphere is V= 4*Pi*R*R*R/3. So for a sphere, the ratio of surface area to volume is given by: S/V = 3/R.
What happens to the volume of a sphere when its radius is doubled? ›Then the radius of sphere is doubled then volume become eight times. Was this answer helpful?
What is the rate of change of the volume of a sphere? ›In a sphere the rate of change of volume is surface area times the rate of change of radius.
What does sphere mean in math? ›A sphere is defined as the set of all points in three-dimensional Euclidean space that are located at a distance. (the "radius") from a given point (the "center"). Twice the radius is called the diameter, and pairs of points on the sphere on opposite sides of a diameter are called antipodes.
Why is it a sphere? ›The Short Answer:
A planet's gravity pulls equally from all sides. Gravity pulls from the center to the edges like the spokes of a bicycle wheel. This makes the overall shape of a planet a sphere, which is a three-dimensional circle.
l = (θ/360) × 2πr or l = (θπr) /180.
What is the formula for area of a sector? ›
The formula for the area of the sector of a circle is 𝜃/360o (𝜋r2) where r is the radius of the circle and 𝜃 is the angle of the sector.
What is the formula for volume of cones? ›The formula for the volume of a cone is V=1/3hπr². Learn how to use this formula to solve an example problem.
What is the definition of sector in math? ›A sector is a region bounded by two radii of a circle and the intercepted arc of the circle. The angle formed by the two radii is called a central angle. A sector with a central angle less than 180° is called a minor sector. A sector with a central angle greater than 180° is called a major sector.
What is section formula in maths? ›In coordinate geometry, Section formula is used to find the ratio in which a line segment is divided by a point internally or externally. It is used to find out the centroid, incenter and excenters of a triangle.
How do you find the area of a circle? ›The area of a circle is pi times the radius squared (A = π r²).
How do you find the central angle of a circle with arc length? ›A central angle is defined as the angle subtended by an arc at the center of a circle. The radius vectors form the arms of the angle. A central angle is calculated using the formula: Central Angle = Arc length(AB) / Radius(OA) = (s × 360°) / 2πr, where 's' is arc length, and 'r' is radius of the circle.
How do you find the area of a sector of a circle without angle? ›Thus the equation of circle is x²+y²=r². The equation of the chord at 'a' distance from center is ax-ry- ar=0 or Y= a/r(x-r). the area of sector can be found by relating it to area of segment where the area of segment is found without the usage of angle made by the chord.
What is segment of a circle examples? ›When you have a slice of pizza and eat the crust, you're having the arc. A segment is the section of a circle enclosed by a chord and an arc. Therefore, those halves of the pizza are segments. If you eat one half, you would have eaten a semicircle (half of a circle), which is the biggest segment of a circle.
How do you find the volume of an object example? ›Whereas the basic formula for the area of a rectangular shape is length × width, the basic formula for volume is length × width × height. How you refer to the different dimensions does not change the calculation: you may, for example, use 'depth' instead of 'height'.
What does it mean to find the volume of a shape? ›Every three-dimensional object occupies some space. This space is measured in terms of its volume. Volume is defined as the space occupied within the boundaries of an object in three-dimensional space. It is also known as the capacity of the object.
What is the definition of volume of cone? ›
The volume of a cone is the amount of space occupied by a cone in a three-dimensional plane. A cone has a circular base, which means the base is made of a radius and diameter.