The best value for each of the characteristics of the parabola are presented in the following table;
The table of best values of parabola characteristics
[tex]\begin{array}{lcccc}\mathbf{Characteristic&\mathbf{r}&\mathbf{ (\pi \cdot r)}&(2 \cdot \pi \cdot \mathbf{r})& ( \pi \cdot \mathbf{r^2})\\\\\mathbf{Base \ of \ parallelogram} & &\checkmark &&\\\\\mathbf{Height \ of \ parallelogram}& \checkmark &&&\\\\\mathbf{Area \ of \ parallelogram}&&&&\checkmark \end{array}}[/tex]
The process of obtaining the above values is shown as follows;
The given parameters are;
To justify the area of a circle, the argument Yvonne wants to use = The dissection argument
The shape to which the reassembled figure tends to approximate = A parallelogram
Description of the of the reassembled figure
The horizontal parallel sides of the parallelogram formed = The arcs of the circle
The other pair of parallel sides = The radius of the circle
The best values of the characteristics of the parallelogram given in the table are therefore found as follows;
(a) The base of the parallelogram = (1/2) the total sum of the arcs in a circle = (1/2) the circumference of the circle = π·r
∴ The base of the parallelogram = π·r
(b) The height of the parallelogram = The perpendicular distance from the tip of one of the sectors to the midpoint of the arc of the sector = The radius of the circle = r
∴ The height of the parallelogram = r
(c) The area of the parallelogram, A = The sum of the area of the sectors
A = (n × θ/360) × π·r²
Where;
n × θ = 360°
Therefore;
A = (n × θ/360) × π·r² = (360/360) × π·r² = π·r²
We have;
The area of the parallelogram, A = π·r²
The table showing the best value of each characteristics of the parallelogram is therefore presented as follows;
[tex]\begin{array}{lcccc}\mathbf{Characteristic&\mathbf{r}&\mathbf{ (\pi \cdot r)}&(2 \cdot \pi \cdot \mathbf{r})& ( \pi \cdot \mathbf{r^2})\\\\\mathbf{Base \ of \ parallelogram} & &\checkmark &&\\\\\mathbf{Height \ of \ parallelogram}& \checkmark &&&\\\\\mathbf{Area \ of \ parallelogram}&&&&\checkmark \end{array}}[/tex]
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