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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

In the topic "mensuration", the foundation focuses on learning

 •  what is measurement standard?

 •  Absolute and Derived Standards

 •  Measurement by superimposition

 •  Measurement by calculation

 •  Measurement by equivalence (infinitesimal and Cavalieri's principle in 2D and 3D. )


The circumference of a circle is calculated by Equivalence (infinitesimal). Let us quickly review the derivation, which was given earlier when discussing Measurement by Equivalence.



click on the content to continue..

Consider the circle given in the figure. The diameter of the circle is `7cm`.area by equivalence Which of the following method helps to compute the circumference of the circle?

  • visualize the circle as a combination of isosceles-triangles and find sum of the bases as circumference
  • visualize the circle as a combination of isosceles-triangles and find sum of the bases as circumference
  • A formula can not be derived for circumference of a plane-figure with curved line

The answer is "visualize the circle as a combination of isosceles-triangles and find sum of the bases as circumference".

Consider finding circumference of a circle spanned with triangles as illustrated in the figure.
approximation
The base of a triangle `bar(AB)` is an approximation to the corresponding arc on the circle `AB`. area by equivalence How to increase the accuracy of the circumference 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 circumference of a circle is calculated by one of many methods involving equivalence and approximation. For now, consider the diameter to be `1cm`.circumference by equivalence The circumference 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 circumference of a circle is a decimal number that does not terminate and does not repeat.

The circumference 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 (a decimal number that does not end or repeat).

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

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

The corresponding triangle for a circle of `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.

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 circumferences of triangles are equal

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

One of the triangles that makes a circle of `1cm` diameter is shown in blue. And one of the triangles that makes a circle of `d cm` diameter is shown in red. These triangles are similar. area by equivalence 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 the triangles from larger circle are `d` times the base of triangles from smaller circle.
This implies that, the circumference of larger circle of `d cm` diameter is `d` times the circumference of the smaller circle.
The circumference of the smaller circle of `1cm` diameter is `pi`.

What does this imply?

  • circumference of the larger circle is `d xx pi`
  • circumference of the larger circle is `d xx pi`
  • it does not imply anything

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

Summarizing the derivation:

Circumference of a circle of `1cm` diameter is computed to be a decimal value that does not end or repeat. That accurate value is represented by the symbol `pi` and approximate value of that is `22/7` and `3.14`.

Circumference of a circle of `d cm` diameter is computed to be `d` times the circumference of the circle of `1cm` diameter.

Thus the generic formula for the circumference of a circle is `pi d` or `2 pi r`.

Circumference of a Circle: :area by equivalence Circumference `= 2 pi r` and `=pi d`

`r` is the radius of the circle

`d` is the diameter of the circle

Solved Exercise Problem:

What is the circumference of a circle of `1/pi cm` diameter?

  • `1cm`
  • `1cm`
  • `pi cm`

The answer is "`1cm`"

                            
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