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Volume of sphere formula

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Mastering the Sphere: Unveiling the Formula for Sphere Volume

Comprehensive Definition, Description, Examples & Rules 

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Introduction

The volume of a sphere measures the space it can occupy. It is an extremely important mathematical formula with application in real-world scenarios, such as determining the size of sports equipment like football, basketball, and tennis balls or calculating the dosage of medicine in tablets.

The Basics of the Sphere

The volume of a sphere represents the amount of space it can contain, which makes it useful in comparing objects with different dimensions. We calculate the volume of a sphere with this formula: 

V= 4/3 πr3

Where,

V is the Volume of the sphere.

r is the radius of the sphere.

This formula is an extremely valuable discovery for us. With the help of this formula, today, we can make multiple important comparisons between different spherical objects of different dimensions, like planets or sports equipment like football, tennis balls, etc. This formula has opened our eyes when it comes to measuring the space in three-dimensional objects and applying that knowledge for a better understanding and functioning of our world. 

The Sphere Volume Formula in Detail

Breakdown of the formula of volume of a sphere

A sphere is formed by placing numerous circular discs on top of each other. The formula of the volume of a sphere is derived by employing integration. The volume of a single disk is expressed as a product of the area of the thin circular disc and its thickness dy. The radius of the disc can be expressed in terms of the vertical dimension(y) using the Pythagorean Theorem. By integrating the limit from -R to R, the total volume of the sphere is obtained. 

Understanding the role of radius in the formula

The radius ‘r’ of a sphere plays a crucial role in the formula of the volume of a sphere 4/3 π r3. As you can see in the formula, the cube of the radius of a sphere is directly proportional to the volume of the sphere; thus, the larger the radius, the larger the volume of the sphere will be. 

Working on the formula 

Let’s understand how to use the formula to find the volume of a sphere with the help of an example. 

Example Question: Find the volume of a sphere whose radius is 7 cm. 

Ans: Given, r= 7 cm

Step 1: Write down the formula

V = 4/3 π r3

Step 2: Substitute the values

V= (4/3).(22/7).(7)3

Step 3: Solve the equation

(4/3).(22/7).(7)3 = 1437.33

Step 4: Write down the final value

V= 1437.33 cm3

Real-World Applications of Volume of a Sphere

There are many real-world applications of the volume of a sphere. A few examples are mentioned.

  1. Manufacturing: The formula of the volume of a sphere is used in manufacturing objects like sports equipment, bubbles, bearings, globes, etc. The formula helps in determining the amount of material required to manufacture a spherical object. 
  2. Chemical Storage: The volume of a sphere formula is used to

 Calculate the amount of harmful chemicals stored in a spherical storage container.

  1. Hot Air Balloons: The volume of a hot air balloon can be calculated using the formula for the volume of a sphere, which helps determine the amount of air in the balloon to prevent leakage. 
  2. Medical Dosage: The volume of the sphere is used in determining the dosage of medicine in medical tablets. 
  3. Planetary Science: The volume of a sphere is also used to calculate the size of planets in space. 

Tips and Tricks for Sphere Volume Calculations

Some tricks to ease your calculation are listed below:

  1. To keep the calculation simple, do not multiply everything and then divide. 
  2. See what you can remove by canceling out. 
  3. Always keep the value of pi 22/7 for easy calculations. 
  4. Try to divide first, then multiply. 

Some mistakes to avoid are listed below:

  1. Make sure that the units are correct.
  2. Avoid multiplication and division errors. 
  3. Be attentive while doing the calculations.

Variations in the Sphere Volume Calculations

Since there are multiple applications of this formula, there are variations to this formula as well. With the sphere volume formula, you can find the volume and dimensions of the sphere, such as the radius or diameter of the sphere. 

Finding the radius or diameter using the sphere volume formula 

When we have the radius of a sphere, we put that value in the sphere volume formula and find out the volume of the sphere. However, if we know the capacity of the sphere beforehand but want to find the dimensions of the sphere, like the radius or the diameter, we use the following: 

r3= ¾* V/π 

Therefore, 

r= ∛(¾*V/π) 

The diameter (d) is two times the radius, hence

d= 2r

The volume of a sphere when we have the diameter

The volume of a sphere can be obtained with the diameter simply by dividing the value of the diameter by 2. 

V= 4/3πr3 

Another method to find the volume of a sphere with the diameter is by substituting the value of r as d/2 in the sphere volume formula. Hence, the new equation becomes: 

V= ⅙πd3

The working of the formula is the same as the original sphere volume formula. 

Relation between the volume of the sphere and the surface area of the sphere

The formula for the surface area (A) of a sphere is provided as

A=4πr2

The relation between the volume of the sphere and the surface area of the sphere is given by this ratio: 2:3. This ratio indicates that for a sphere of any dimension with a surface area A, the volume of that sphere V will be ⅔ times the surface area. 

V=⅔ A

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Key Takeaways

  1. The volume of a sphere measures the capacity/space within the sphere.

  2. The volume of a sphere (V)  when the radius is given by  4/3 π r3 unit3.

  3. The volume of a sphere (V) when the diameter is given by V= ⅙πd3 unit3.

  4. The volume of a sphere and the cube of the radius are directly proportional.

  5. The volume of a sphere is ⅔ times the surface area of the sphere.

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Frequently Asked Questions

The volume of the sphere is given by 4/3 π r3.

Yes, there are different ways to calculate the sphere volume concerning diameter and surface area. The sphere volume concerning diameter is given by  V= ⅙πd3. The sphere volume concerning radius is given by ⅔* 4πr2.

You can efficiently calculate the sphere volume by being attentive and not blindly multiplying all the values.

Yes, the sphere volume is related to surface area; V=⅔ A.

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2D Shapes2cosacosb Formula30-60-90 Formulas3D ShapesAbsolute Value FormulaAcute AngleAcute Angle triangleAdditionAlgebra FormulasAlgebra of MatricesAlgebraic EquationsAlgebraic ExpressionsAngle FormulaAnnulusAnova FormulaAnti-derivative FormulaAntiderivative FormulaApplication of DerivativesApplications of IntegrationArc Length FormulaArccot FormulaArctan FormulaArea Formula for QuadrilateralsArea FormulasArea Of A Sector Of A Circle FormulaArea Of An Octagon FormulaArea Of Isosceles TriangleArea Of ShapesArea Under the Curve FormulaArea of RectangleArea of Regular Polygon FormulaArea of TriangleArea of a Circle FormulaArea of a Pentagon FormulaArea of a Square FormulaArea of a Trapezoid FormulaArithmetic Mean FormulaArithmetic ProgressionsArithmetic Sequence Recursive FormulaArithmetic and Geometric ProgressionAscending OrderAssociative Property FormulaAsymptote FormulaAverage Deviation FormulaAverage Rate of Change FormulaAveragesAxioms Of ProbabilityAxis of Symmetry FormulaBasic Math FormulasBasics Of AlgebraBinary FormulaBinomial Probability FormulaBinomial distributionBodmas RuleBoolean AlgebraBusiness MathematicsCalculusCelsius FormulaCentral Angle of a Circle FormulaCentral Limit Theorem FormulaCentroid of a Trapezoid FormulaChain RuleChain Rule FormulaChange of Base FormulaChi Square FormulaCirclesCircumference FormulaCoefficient of Determination FormulaCoefficient of Variation FormulaCofactor FormulaComplex numbersCompound Interest FormulaConditional Probability FormulaConeConfidence Interval FormulaCongruence of TrianglesCorrelation Coefficient FormulaCos Double Angle FormulaCos Square theta FormulaCos Theta FormulaCosec Cot FormulaCosecant FormulaCosine FormulaCovariance FormulaCubeCylinderDiscover the world of MathsEllipseEquilateral triangleEuler’s formulaEven numbersExponentsFractionHeptagonHyperbolaIntegersIntegrationIntegration by partsLinesLocusMatricesNatural numbersNumber lineParallelogramPercentage formulaPerimeterPolygonPolynomialsPrismProbabilityPyramidPythagoras theoremScalene triangleSetsSlope formulaSolid shapesSphereSquareStandard deviation formulaSubtractionSymmetryTimeTrianglesTrigonometry formulaTypes of anglesValue of PiVectorVolume formulasVolume of a coneVolume of sphere formulaWhole numbers
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