**Volume of spheres (Cavalieri´s principle)**

Calculating the volume of a sphere is possible using an appropriate cylinder and cone.

**Mathematics**

**Keywords**

volume of sphere, Cavalieri´s principle, calculating volume, solids, sphere, mathematics

**Related items**

### Scenes

### Demonstration of Cavalieri´s principle

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### Cavalieri´s principle

Take two solids and place them on one plane. Intersect them with two planes parallel to their base and examine the two solids and their cross-sections according to the following properties:

- The areas of their bases are equal.

- The areas of all cross sections parallel to their bases are equal.

- The heights of the two solids are equal.

If all of these are true, the volumes of the two solids are equal.

Cavalieri´s principle helps a lot in calculating the **volume of spheres**. Without it, higher mathematical methods would need to be applied in order to come to a result.

Let´s consider a hemisphere of radius **r** with its cross-section, and a cylinder which lies in the same plane. The radius of the circular base and the height of the cylinder are **r**. Let´s cut from the cylinder an upturned cone, with both radius and height **r**. In the animation, these solids are shown together with their mirror images relative to the plane. The areas of the bases of the two solids are equal.

When examining their cross-sections parallel to this plane we have to calculate the area of the cross-sections, which is at a height **h**.

In the case of the sphere the cross-section is a **circle**. Due to the Pythagorean theorem, the square of the circle's **radius** equals to **r² - h²** , thus its **area **is

In the case of the other solid the cross-section is an annulus with an external radius **r** and an internal radius **h**. Its area is

**In other words, given two solids, the surface areas of their cross-sections parallel to their base are equal.**

Due to the formation of the two solids, **the heights of the two solids are equal.**

All the conditions of Cavalieri´s principle are met, therefore **the two solids have an equal volume.**

### Animation

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