*To measure a length, area, or volume one of the following methods is used. • Measurement by Superimposition • Measurement by Calculation • Measurement by Equivalence We studied about "measurement by superimposition" and "measurement by calculation". Concept of " Measurement by Equivalence" is easy to understand. Three methods are used to find equivalence. These three methods are little involved in explanation and are hard to understand. These are essential to solve area of shapes of curves or volume with curved surfaces. • Equivalence by infinitesimal pieces • Equivalence by Cavalieri's Principle (2D) • Equivalence by Cavalieri's Principle (3D)*

In this topic, Equivalence by Cavalieri's Principle (3D) is explained with an example.

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Consider the solid given in the figure. It consists of four flat surfaces in the front, top, back, and bottom. And a pyramidal projection on the right. A similar pyramidal hollow-space on the left.

Spend some time to visualize this.

Consider finding volume of the solid given in the figure. The solid is given in orange on the left. Another solid, a cuboid, is considered. The cuboid is shown in blue.

The two figures are aligned and a plane `A` intersecting them is drawn. The intersecting plane creates two plane-figures in the two solids at the cross-section. The two plane-figures are given as `P` and `Q`.

It is noted that the area of plane-figures are equal (by Cavalieri's principle in 2D).

As per Cavalieri's principle, if the areas of the cross-sections are equal for any plane parallel to the one shown, then the volume of the two solids are equal.

What does this imply?

- volume of the solid is half that of the cuboid
- volume of the solid equals the volume of the cuboid
- volume of the solid equals the volume of the cuboid

The answer is "volume of the solid equals the volume of the cuboid"

**Cavalieri's Principle in 3D** : For a given two solids, a plane intersecting the solids is considered. The area of the intersecting cross-sections `P` and `Q` are considered. If the areas of intersecting cross-sections are equal for all planes parallel to the intersecting plane, then the volumes of the two solids are equal.

*slide-show version coming soon*