*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 (2D) is explained with an example.

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Consider the shape given in the figure. It consists of two straight lines and two curved lines. Which of the following help to find the area of the figure?

- approximate area can be computed by superimposing unit-squares and counting them
- accurate area can be computed by some equivalence
- both the above
- both the above

The answer is "both the above". This problem is used to illustrate and understand Cavalieri's principle.

Consider finding area of the shape given in the figure. The shape is given in orange. Another shape, a rectangle, is considered. The rectangle is shown in blue.

The two figures are aligned and a line intersecting them is drawn. The point of intersections are shown as `A`, `B`, `C`, and `D`.

It is noted that the length of `bar(AB)` equals the length of `bar(CD)` for any position along vertical direction. These two are shown as line segments `P` and `Q`.

As per Cavalieri's principle, if the length of the line-segments are equal for any line parallel to the one shown, then the area of the two shapes are equal.

What does this imply?

- area of the shape is twice that of the rectangle
- area of the shape equals the area of the rectangle
- area of the shape equals the area of the rectangle

The answer is "area of the shape equals the area of the rectangle"

**Cavalieri's Principle in 2D** : For a given two shapes on a plane, a line intersecting the shapes is considered. The length of the intersecting line segments `bar(AB)` and `bar(CD)` are considered. If the length of the intersecting line-segments are equal for all parallel lines to the intersecting line, then the areas of the two shapes are equal.

*slide-show version coming soon*