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Cavalieri's Principle

Geometry is a special section of mathematics. It studies about the shapes, areas, volumes and many other calculations on geometrical figures. Geometry is full of theorems and principles. Cavalieri's principle is one of those. It was given by the mathematician Bonaventura Cavalieri after whose name this principle is called. Originally, Cavalieri's principle was termed as the method of indivisibles.
Cavalieri's principle was used mainly to find the volumes of the three-dimensional shapes, such as: sphere, pyramid, cone, cylinder etc. According to this principle, a plane figure is thought as if it is made up of infinite number of lines having infinitesimal width; i.e. a plane figure can be made up of infinite parallelograms of negligible breadth.Similarly, a three-dimensional figure is thought of made out of infinite planes of infinitesimal width. In this way, the area or volume can be found by applying the formula for the sum. So, let us go ahead in this page and learn about Cavalieri's principle and its application in detail.

 

Definition

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Cavalieri's principle states that:
In two-dimensions: If in two plane figures, there are sections made by two parallel lines at the same distance in both the figures. And if each line parallel to these lines intersects both figures in the line segments of same length, then the two plane figures are said to have same areas.

In three-dimensional: If in two solid figures, the sections are made by two parallel planes at the same distance in both of them. If each plane parallel to these plane intersects both solids in the planes of same area, then the two solid figures will be having same volumes.
Lets understand this in simpler words. If two 2-dimensional figures are constructed between the same parallel lines and if straight lines in them are drawn at the equal distance from these parallel lines, then the included parts of any of these lines are equal and the areas of plane figures are also equal.

Also, if two 3-dimensional figures are constructed between the same parallel planes and if any planes are drawn in them at a same distance from these parallel planes, Then the included plane figures so drawn are equal and the solid figures have equal volumes. We can say that if in two solid figures of same altitude, the sections that are made by means of planes parallel to and locating at a same distance from their bases, are equal, and hence volumes of the two given solids are also equal.

Applications

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Cavalieri's principle has many applications in mathematics. It is used in finding the areas and volumes of various figures. It is utilized in estimating the exact area or volumes of a non geometrical if it is in same parallels and on a base with same dimensions.

Cavalieri's principle is not only used in mathematics, but also in many practical methodologies. One of most common application of this principle is in medical sciences. Cavalieri's principle is used in stereological analysis of human organs using unbiased procedures, such as - unbiased estimation of the volume of lung with the help of Cavalieri's principle.

Example

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There are various examples of Cavalieri's principle. Few of those are illustrated below:
In Two-Dimensions
1) Triangles:

Have a look at the following diagram:
Triangles

If two triangles are formed on base of equal length and are between same parallels; i.e have same height, then the triangles are of equal area. This is a very common example of Cavalleri's principle.

2) Parallelograms:
One more most common example for two-dimensional figures is that if two parallelograms are between the same parallel lines and are made up on base of equal length, then their area will be same, as shown in the following figure :
Parallelogram

3) Other Shapes:
Same principle is applied on other arbitrary shapes too. Refer the diagram given below :
Circles

Here, there is no common base, but every line that is drawn between two parallel lines in one shape must be equal to another line situated at the same height in other shape.

In Three Dimensions:
1) Cylinders:
Have a look at two cylinders below. These cylinder have same height and same area of base. Then, these are supposed to have same volumes.
Cylinders

Therefore, we can say that the volume of a right circular cylinder is same as the volume of any other cylinder with same area of base and height.

2) Prisms:
In the following diagram, two prisms are shown which are constructed between two parallel lines and any plane drawn between these parallels at same height have equal area. Thus, the volumes of these prisms are equal.
Prisms

3) Pyramids:
Have a look at the following two pyramids. They have same height. Every plane in one pyramid at same height in the other are equal to one another. Then the volumes of two pyramids are equal.
Pyramids


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