__maths__>__Mensuration : Length, Area, and Volume__>__Measurement Methods__### Equivalence by Infinitesimal Pieces

*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. The following three methods are explained later in the course. • Equivalence by infinitesimal pieces • Equivalence by Cavalieri's Principle (2D) • Equivalence by Cavalieri's Principle (3D)*

In this topic, Equivalence by infinitesimal pieces is explained with an example.

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Consider the circle given in the figure. The diameter of the circle is `7cm`. Can the perimeter of the circle be accurately calculated by superimposition?

- accurate value is possible
- only approximate value is possible
- only approximate value is possible

The answer is "only approximate value is possible". As the length of unit-line is made finer, the accuracy would increase.

Consider finding perimeter of a circle of diameter `7cm`. The figure illustrates the problem. The circle is overlapped with triangles. Two sides of the triangles equal to the radius of the circle, and the third side (referred as base) is along the curved line of the given circle. The figure shows `10` such triangles occupying a quarter of the circle. It is visualized that the three other quarters are also filled with triangles.

The length of the base of the triangles are added. Will it be accurately equal to the perimeter of the circle?

- yes, it is accurate
- No, it is approximate
- No, it is approximate

The answer is "it is approximate". The base of triangles are straight lines but the section of circle is curved. The length of the base (which lies on the circle) will be little less than the length of the curved section.

Consider finding perimeter of the circle with triangles as illustrated in the figure. How to increase the accuracy of the perimeter value?

- accuracy cannot be increased
- if the triangle is made thinner (angle between equal sides is smaller), then the accuracy increases
- if the triangle is made thinner (angle between equal sides is smaller), then the accuracy increases

The answer is "if the triangle is made thinner, then the accuracy increases".

The perimeter of a circle is calculated by one of many methods involving equivalence and approximation. For now, consider the diameter to be `1cm`. The perimeter of the circle is found to be `3.14159cdots` by approximating the circle to infinitesimally small triangular pieces.

Remember we studied that length of a line-segment can be measured finer and finer to an accurate value -- a decimal number that does not terminate and does not repeat. Similarly, the perimeter of a circle is a decimal number that does not terminate and does not repeat.

The perimeter of a circle of diameter `1cm` is commonly approximated to a value `22/7` or `3.14`.

To represent the accurate decimal value, the number is represented with the symbol `pi`. It is an irrational number or a decimal number that does not end or repeat.

The perimeter of a circle with diameter `1cm` is computed as a value represented with `pi`.

What will be the perimeter of a circle with diameter `2cm` or `20 cm`? Will there be more numbers like `pi`? The question is answered with some geometrical calculations. Consider one piece of the triangle that was used to calculate perimeter of a circle of diameter `1cm`. The triangle is shown in blue color in the figure.

The corresponding triangle for a perimeter `d cm` is shown in red. Note that the triangles overlap and the blue triangle is shown on top of the red one.

The corresponding angles of these two triangles are equal and thus the triangles are similar triangles.

Which of the following is a property of similar triangles?

- the ratio of corresponding sides are constant
- the ratio of corresponding sides are constant
- the perimeters of triangles are equal

The answer is "the ratio of corresponding sides are constant"

One of the triangles making a circle of `1cm` diameter is shown in blue. And one of the triangles making a circle of `d cm` diameter is shown in red. These triangles are similar. The property of similar triangles is that, the ratio of the sides are constant.

The isosceles sides of blue triangle are `1//2 cm` and the base is `x`.

The isosceles sides of red triangles are `d//2 cm` and the base is `y`.

As per property of similar triangles,

`1//2 -: 1//d = x -:y`

`-> y=d xx x`

This means, the base of triangles from larger circle are `d` times the base of triangles from smaller circle.

This implies that, the perimeter of larger circle of `d cm` diameter is `d` times the perimeter of the smaller circle.

The perimeter of the smaller circle of `1cm` diameter is `pi`.

What does this imply?

- perimeter of the larger circle is `d xx pi`
- perimeter of the larger circle is `d xx pi`
- it does not imply anything

The answer is "perimeter of the larger circle is `d xx pi`".

**Equivalence by Infinitesimal Pieces** : Length, Area, and Volume can be measured by equivalence with infinitesimal pieces. An example of finding perimeter of a circle is illustrated in the figure. The circle is spanned with isosceles triangles.

Approximate value of the perimeter of the circle equals sum of the bases of the triangles.

Accurate value of the perimeter of the circle is calculated by making the triangles infinitesimal.

*slide-show version coming soon*