Question

Find the length of the perimeter of a regular decagon which surrounds a circle of radius $ 12cm $

Answer 1

Hint: Polygon with ten sides is known as the decagon. It has ten vertices and ten edges. The perimeter of the regular decagon is the sum of all the ten sides of the polygon. First we will find length using formula $ l = 2r\tan \dfrac{{180^\circ }}{n} $ and then substitute the value of length in the formula of the perimeter, $ p = 10l $

Complete step-by-step answer:
Given that the decagon surrounds the circle.
Therefore radius of the in-circle, $ r = 12cm $
The side of the regular decagon is $ n = 10 $
Let, length of the regular decagon be $ = l $
Now, according to the formula –
 $ l = 2r\tan \dfrac{{180^\circ }}{n} $
Place the known values in the above equation –
 $ l = 2 \times 12\tan \dfrac{{180^\circ }}{{10}} $
Simplify the above equation –
 $ l = 24\tan 18^\circ $
For the values of $ \tan 18^\circ $ refer log book. Just refer the value in the natural tangent in the regular log table. We get $ \tan 18^\circ = 0.32492 $
Place it in the above equation.
 $
\Rightarrow l = 24 \times 0.32492 \\
\Rightarrow l = 7.798cm \\
  $
Now, the perimeter of the decagon is the sum of all its ten sides.
 $ \therefore $ Perimeter $ = 10l $
Substitute the value of length in the above equation –
 $ \therefore $ Perimeter $ = 10 \times 7.798 $
 $ \therefore $ Perimeter $ = 77.98cm $
Hence, the required answer - the length of the perimeter of a regular decagon which surrounds a circle of radius $ 12cm $ is $ 77.98cm $

Note: The self-intersecting regular decagon is known as decagram. Use of a logarithmic table should be done very carefully. Use scale for the precise numbers and to stay in the same required row. We have many values of the functions; refer to the page of the required trigonometric function properly.
The sum of all the interior angles of the regular decagon is $ 1440^\circ $