# Volume of Sphere

**What is the Sphere?**

Geometrically, a sphere is a set of all points in space that lie at the same fixed distance from a reference point. The reference point is called the *center of the sphere* and a fixed distance taken from the center is called the *radius* of the sphere. A sphere can also result if you rotate a circle about any of its diameters.

**Also Read: ****Median of a Triangle**

**The volume of a Sphere**

**If we assume the radius of a sphere to be ****r****, then the volume of a sphere is given by the formula: **

**V=43r3**

However, the volume of the **sphere** was derived over two thousand years ago by the Greek mathematician and **philosopher- Archimedes**. At the time, calculating the volume of straight **3- D figures** like a cube or cuboid was fairly easy. But calculating the areas and volumes of curved figures, such as the volume of the sphere, was quite a challenge.

If we try to derive a formula for the volume of a sphere today, we will need a deeper understanding of calculus. Interestingly, calculus was derived only in the 17^{th} century by **Isaac Newton**. This means that Archimedes did something similar to integral calculus, long before it was discovered, to find the volume of a sphere.

**Also Read: ****Mathematicians**

**Discovery**

The main challenge in finding the volume of a sphere at the time was that the surface changes direction at every point. This is how Archimedes worked with it: –

Imagine cutting a sphere in half, so you have a hemisphere. Now, imagine this hemisphere inscribed perfectly inside a cylinder.

In the first slice, the part blue part representing the sphere will be infinitesimally small. However, as we keep moving downwards, the hemisphere’s cross-section starts to grow until the green region disappears completely in the last slice.

Now keep in mind, there can be as many slices as you like, and they can be as thin as you like. Now if we multiply the areas of all these green rings with their depths, we will find the total volume of all these green rings stacked up together (excluding the blue part).

Archimedes found that this volume of the green part added up to the volume of a cone, with base radius and height being the same as that of the cylinder.

From this, he inferred that the volume of the hemisphere was equal to the volume of cylinder minus the volume of a cone having the same base radius and height.

Now, the volume of the cylinder was already known to be **r2h** and the formula for the volume of the cone was known to be **13r2h**.

**Also Read: **Number Series

**From this, we get the volume of the hemisphere as**

**r2h – 13r2h = 23r2h**

In this case, the radius and height are the same, and equal to r.

**So, the volume of the hemisphere is 23r3.**

The volume of a sphere, therefore, will be double the volume of the hemisphere, i.e.,

**V=43r3**

**FAQs**

**✅ What is a formula for the volume of a sphere?**

The **formula for the volume of a sphere** is V = 4/3 πr³

**✅ Why is the sphere volume formula?**

The cylinder **volume** is πR3, the cone is a third that, so the hemisphere **volume** is 23πR3. Thus the **sphere** of radius R has **volume** 43πR3.

**✅ What is a formula of a cylinder?**

The **formula** for the volume of a **cylinder** is V=Bh or V=πr2h

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