Radius of a Sphere Calculator

Created by Dominik Czernia, PhD
Reviewed by Bogna Szyk and Adena Benn
Last updated: Nov 21, 2022

Our radius of a sphere calculator is a perfect tool that can estimate every parameter of a sphere from just one other quantity. However, we designed it to support the computation of a sphere's radius. Try this calculator now by entering one of the chosen parameters into the appropriate field or read on and learn how to find the radius of a sphere. In the following text, we have also presented four different radii of a sphere formulas.

A sphere is a perfectly round geometrical object in 3D space. The points on its surface are equally distant from the center. It is an analog to a circle in 2D space. The radius of a sphere calculator uses five variables that can completely describe any sphere:

How to find the radius of a sphere? - a picture
  • rr - radius of a sphere;
  • dd - diameter of a sphere;
  • VV - volume of a sphere;
  • AA - area a sphere; and
  • A/VA / V - surface to volume ratio of a sphere.

A sphere is a special object because it has the lowest surface to volume ratio among all other closed surfaces with a given volume. Also, here we can find the analogy to the circle, which encloses the largest area with a given perimeter. This radius of a sphere calculator, as the name suggests, contains information dedicated mostly to the radius of a sphere. For more general information about spheres, check out our sphere calc!

There is an object called hemisphere that you can construct from any sphere you want. You just need to divide a sphere into two equal parts. The description of a hemisphere is a little bit more complicated compared to the full sphere, but it is possible. If you want to learn more about that kind of object, check out our area of a hemisphere calculator and volume of a hemisphere calculator.

What's the radius of a sphere formula?

How to find the radius of a sphere? Actually, there are many varied answers to that question because it depends on what we know about a specific sphere. Below, we have provided an exhaustive set of the radius of a sphere formula:

  1. Given diameter: r=d/2r = d / 2,
  2. Given area: r=A/(4×π)r = \sqrt{A / (4 \times π)},
  3. Given volume: r=3×V/(4×π)3r = \sqrt[3]{3 \times V / (4 \times π)},
  4. Given surface to volume ratio: r=3/(A/V)r = 3 / (A/V).

Our radius of a sphere calculator uses all of the above equations simultaneously, so you need to enter just one chosen quantity. Moreover, you can freely change the units (SI and imperial units). Check out our length conversion to learn how to switch between different units of length!

Derivation of the above radius of sphere formulas is, in fact, straightforward. You need to make some algebraic transformations using the following basic equations:

  1. Diameter of a sphere: d=2×rd = 2\times r;
  2. Surface area of a sphere: A=4×π×r2A = 4 \times π \times r^2;
  3. Volume of a sphere: V=4/3×π×r3V = 4/3 \times π \times r^3; and
  4. Surface to volume ratio of a sphere: A/V=3/rA / V = 3 / r.

How to find the radius of a sphere?

This question seems very easy at first. You only have to find the center of the sphere and measure the distance to any point on its surface. However, how can you find this center in the real, physical sphere, especially when it's closed? Let's look on a two of our suggestions:

  1. Find the diameter. You can use two parallel planes and put the sphere between them. The distance between those plates equals the diameter of a sphere.
  2. Find the volume. Prepare a graduated cylindrical container filled with water. Put a sphere into this cylinder, then bring out the sphere and measure the volume of displaced water (you can use our volume of a cylinder calculator to do so). If you want to see how objects behave in liquids, try our buoyancy calculator now!

We didn't write about finding the surface area of the sphere first because it is much harder to do it (compared to finding the radius).

It is worth mentioning that spheres can often simplify various problems in physics. That's why they are widely used in this field, e.g., to model spherical capacitors or atoms of a gas.

FAQ

How do I calculate the radius of a sphere given the volume?

To calculate the radius of a sphere given the volume:

  1. Multiply the volume by three.
  2. Divide the result by four times pi.
  3. Find the cube root of the result from Step 2.
  4. The result is your sphere's radius!

What is the radius of a sphere with area 50 cm²?

1.99 cm. To calculate the radius r of a sphere given the surface area (A), rearrange the formula:
A = 4 × π × r²
To isolate the radius:
r = √[A / (4 × π)]
Notice how the area of a sphere is exactly four times the area of the circle with the same radius!

How do I calculate the radius of Earth from its volume?

If you approximate the Earth as an ideal sphere (it is, in fact, a geoid) and you know the volume, you can calculate its radius. The numbers will be big, though!
The volume V of Earth is:
1,083,206,916,846 km³
Apply the inverse formula for the volume of a sphere to find the radius:
r = ³√[3 × V / (4 × π)] = 6371 km.
This is not a bad approximation, as the radius of Earth actually ranges from 6357-6378 km!

How do I measure the radius of a sphere?

Measuring the radius of a sphere is not that easy. You can wrap a string around it and hope you got the great circle, or you can be a bit more creative:

  1. Fill an open container (of which you know the surface area) with water.
  2. Mark the water level.
  3. Submerge the ball (using your hand) and measure the new water level.
  4. Submerge only your hand as much as before and measure the water level again.
  5. Compute the volume of the water displaced by the ball only by subtracting the displacement of just your hand from the displacement of both objects.
  6. Apply the formula r = ³√[3 × V / (4 × π)] to find the radius r.
Dominik Czernia, PhD
How to find the radius of a sphere? - picture
Diameter
in
Circumference
in
Surface area
in²
Volume
cu in
Radius
Radius
in
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