Question

If the perimeter of a semi – circle protractor is 108cm, find the diameter of the protractor. (Take \[\pi =\dfrac{22}{7}\])

Answer 1

Hint: Find the perimeter of semi – circular protractor, which is the sum of the perimeter of semi – circle and diameter. Equate this perimeter of semi – circular protractor to our given perimeter 108cm and find the diameter.

Complete step-by-step answer:
Let us consider the radius of the semi – circular protractor as r. Now for a circle the diameter of the circle is twice the radius, which we can write as, d = 2r.
Now let us find the perimeter of the semi – circle protractor.
We know that the perimeter of semi – circle = \[\pi r\].
Thus the perimeter of the semi – circular protractor = d + \[\pi r\], put d =2r
Perimeter = 2r + \[\pi r\]
\[\therefore \] Perimeter of the semi – circular protractor = 2r + \[\pi r\] = r (2 + \[\pi \]).
Now put, \[\pi =\dfrac{22}{7}\].
\[\therefore \] Perimeter = \[r\left[ 2+\dfrac{22}{7} \right]=r\left[ \dfrac{14+22}{7} \right]\]
\[\therefore \] Perimeter = \[r\left( \dfrac{36}{7} \right)\]
We have been given the perimeter of the semi – circular protractor as 108cm.
Now let us equate both.
Perimeter of semi – circular protractor = \[r\left( \dfrac{36}{7} \right)\].
\[\begin{align}
  & \Rightarrow 108=\dfrac{36r}{7} \\
 & \therefore 36r=108\times 7 \\
\end{align}\]
\[r=\dfrac{108\times 7}{36}=3\times 7=21\]cm
Thus we got the radius of the semi – circle as 21cm.
Similarly, diameter, d = 2r = 2 \[\times \] 21 = 42cm.
Hence the radius = 21cm and diameter = 42cm.
 \[\therefore \] Diameter of the semi – circular protractor = 42cm.


Note: The perimeter of a semicircle is half of the circle. The perimeter of circle = \[2\pi r\].
\[\therefore \] Perimeter of semi – circle = half of perimeter of circle = \[\dfrac{1}{2}\times \left( 2\pi r \right)=\pi r\].
The possible mistake is that students might forget to add the diameter to the perimeter of the semi – circle and this will not be complete. This will lead to the wrong answer as well.