How to Find the Volume of a Sphere

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Learn to calculate the volume of a sphere with one simple formula
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In geometric terms, a sphere is defined as a set of points that are a given distance from a given point, and that point is the center. The volume of a sphere is how much space is inside it, and calculating this volume is easy! All you need to do is find the sphere’s radius, then plug it into a simple formula, V = ⁴⁄₃πr³.

Volume of a Sphere Formula

Calculate the volume of a sphere with the formula V = ⁴⁄₃πr³ , where V = volume and r = radius. Find the radius by halving the diameter, or by using the formula r=c/(2π), where c = circumference.

Section 1 of 3:

Solving for the Volume of a Sphere

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  1. This is the equation: V = ⁴⁄₃πr³ . In this equation, "V" represents volume and "r" represents the radius of the sphere. [1]
    • Sometimes, you’ll be given each of these measurements, and then you can just plug them into the equation. Other times, you may have to find them yourself before you can calculate the sphere’s volume.
  2. If you're given the radius, then plug it into the formula and move on to the next step. If you're given the diameter, then you can just divide it by two to find the radius . [2] Let's say the radius we're working with is 1 inch (2.5 cm).
    • Otherwise, see the section below on finding the radius of a circle.
    • Example 2: We’ll also work with a second example, to help you understand. For this second example, we’ll say the radius is 2 inches.
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  3. To cube the radius, simply multiply it by itself three times, or raise it to the third power. For example, 1 inch 3 is really just , since 1 multiplied by itself any number of times will be 1. Then, plug the cubed radius into the original equation for calculating the volume of a sphere, V = ⁴⁄₃πr³ . [3] So, V = ⁴⁄₃π x 1 .
    • Example 2: If the radius were 2 inches, for example, then to cube it, you would find 2 3 , which is 2 x 2 x 2, which equals 8.
    • You'll reintroduce the unit of measurement, which in this case is inches, when you state your final answer.
  4. Now that you've plugged r 3 , or 1, into the equation, multiply this result by 4/3 to start actually solving the equation V = ⁴⁄₃πr³ . 4/3 x 1 = 4/3. Now, the equation will read V = ⁴⁄₃ x π x 1, or V = ⁴⁄₃π. [4]
    • Example 2: In our other example, we found that the cubed radius (2 3 ) was 8. Multiply 8 by 4/3 to get 32/3.
    • We multiply it by 4/3 first, instead of by π first, because it’s easier to multiply things by 4/3 than by π, and since it’s a simple multiplication problem, it doesn’t matter which number we multiply by first—our result will be the same.
  5. This is the last step to finding the volume of a sphere. You can leave π as it is for the most accurate answer, stating the final answer as V = ⁴⁄₃π. [5] Or, plug π into your calculator and multiply its value by 4/3. The value of π (approximately 3.14159) x 4/3 = 4.1887, which can be rounded to 4.19. So, the volume of a sphere with the radius of 1 is 4.19 in. 3
    • Remember to state your final answer in cubic units!
    • Example 2: Earlier, we found that 4/3 * 8 = 32/3. Now, we multiply 32/3 by π, which is about 33.5 (rounded), so our answer is ~33.5 inches 3 .
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Section 2 of 3:

Finding Radius

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  1. If you’re given the diameter, finding the radius is easy: it’s just the diameter divided by 2! [6] So if the diameter of the circle is 5 cm, for instance, then cm.
  2. Some math problems will ask you to solve for the volume of a sphere, but will only give you its circumference. In this case, we first use the formula r=c/(2π) to find the radius , where r = radius and c = circumference. Then, once we’ve found the radius, we can plug it into the volume equation like usual.
    • For example, if the circumference of the circle is 4 inches, then r = 4 / (2π). First, divide 4 by 2 to get 2, so we have the equation r = 2 / π. Then, 2 / π = 0.64 (rounded), so r = 0.64 inches.
    • Or, we can do it another way by first multiplying 2 by π, which is about 6.28 (rounded), then divide 4 by 6.28, which equals 0.64 inches.
  3. If you’re given the surface area of the sphere and asked to find the volume, then you’ll have to backtrack a bit to first find the radius, then plug the radius into the volume equation above. To find the radius when you’re given the surface area, the equation is r = √(A / [4 * π]), where r = radius and A = surface area of the sphere. [7] .
    • For example, if the surface area is 2 inches, then our equation for radius is r = √(2 / [4 * π]). 4 * π = 12.57 (rounded). 2 / 12.57 = 0.16 (rounded). Then, use a calculator to find the square root of 0.16, which is 0.4, so our radius here is 0.4 inches.
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Section 3 of 3:

Why does this volume formula work?

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  1. This formula can be explained with a little geometry. You’re probably asking, “Where did the 4/3 come from?” To get a precise answer, you’ll have to study calculus, where a long, long formula involving derivations and functions can prove the equation. Or, we can look at a slightly similar Greek proof, which goes like this: [8]
    • Say we have a sphere, and we want to know the volume of it. First, we pretend there’s a cylinder surrounding the sphere that shares the sphere’s diameter and height. We know how to find the volume of the cylinder, which is the equation: Area of the base of the cylinder x height, or πr 2 x height.
    • Now, say we have an upside-down cone, and the point of the cone is the center of the sphere, while the base of the cone is the ceiling of the cylinder. We also know how to find the volume of a cone with the equation (1/3)(πr 3 ).
    • At this point, we could say that, if we were to take a flat cross-section of these overlapping shapes, then the area of the sphere within that cross-section would be equal to the area of the cylinder minus the area of the cone. Then, to find the total volume of the inner sphere, we’d keep taking cross sections this way, subtracting the cone’s area from the cylinder’s, then adding up all our results.
    • To describe that process as a sentence, we’d say that the volume of the cylinder (πr 3 ) minus the volume of the cone (1/3)(πr 3 ) equals the volume of the sphere, multiplied times 2, since we actually need 2 cones to cover the entire height of the sphere, so we’d do the process twice.
    • To describe the above sentence as a mathematical equation, we say (πr 3 ) - (1/3)(πr 3 ) = (2/3)(πr 3 )(2) = (4/3)(πr 3 ) .

Practice Problems and Answers

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  • Question
    How do I calculate the volume of a sphere if neither the radius nor the diameter is given?
    Donagan
    Top Answerer
    If you know the surface area, solve the area formula for the radius, and use that to find the volume. Without the radius, you can't determine the volume.
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    Not Helpful 29 Helpful 82
  • Question
    Why is the formula not V = pi x R squared x H?
    Diante Watts
    Community Answer
    Height is not included when measuring spheres, since usually they are congruent in all directions; it wouldn't be a necessity, because the radius is included.
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    Thank you for your feedback.
    If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
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  • Question
    How can I tell the volume with only the diameter given?
    Community Answer
    Divide the diameter by 2, giving you the radius and continue from there.
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      References

      1. ↑ https://www.purplemath.com/modules/geoform.htm
      2. ↑ https://www.mathsisfun.com/definitions/diameter.html
      3. ↑ https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_II_(Illustrative_Mathematics_-_Grade_8)/05%3A_Functions_and_Volume/5.5%3A_Dimensions_and_Spheres/5.5.4%3A_The_Volume_of_a_Sphere
      4. ↑ https://www.cuemath.com/measurement/volume-of-sphere/
      5. ↑ https://www.mathsisfun.com/geometry/sphere-volume-area.html
      6. ↑ https://www.mathsisfun.com/definitions/diameter.html
      7. ↑ https://www.web-formulas.com/Math_Formulas/Geometry_Surface_of_Sphere.aspx
      8. ↑ http://mathcentral.uregina.ca/QQ/database/QQ.09.01/rahul1.html

      About This Article

      Article Summary X

      To calculate the volume of a sphere, use the formula v = ⁴⁄₃πr³, where r is the radius of the sphere. If you don't have the radius, you can find it by dividing the diameter by 2. Once you have the radius, plug it into the formula and solve to find the volume. For more tips, including examples you can use for practice, read on!

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