Question

In a polygon, there are 5 right angles and the remaining angles are equal to $$195^{\circ}$$ each. Find the number of sides in the polygon.

Answer 1

- Hint: In this question it is given that in a polygon, there are 5 right angles and the remaining angles are equal to $$195^{\circ}$$ each. We have to find the number of sides in the polygon. So to find the solution we need to know that if a polygon has ‘n’ number of sides then the summation of all of its interior angle is $$\left( n-2\right) 180^{\circ }$$ , so by using this formula we will get the value of n for the given question.

Complete step-by-step solution -
Let us consider ‘n’ to be the total number of sides.
Therefore, here it is given that 5 angles of a polygon is $$90^{\circ}$$ [i.e, right angle] and the remaining $$195^{\circ}$$.
Then the summation of all the angles will be $$5\times 90^{\circ }+\left( n-5\right) 195^{\circ }$$.........(2)
Also from formula (1) we can say that for any polygon the summation of all its interior sides is $$\left( n-2\right) 180^{\circ }$$.......(3)
So from (2) and (3) we can write,
$$\left( n-2\right) \times 180^{\circ }=5\times 90^{\circ }+\left( n-5\right) \times 195^{\circ }$$
$$\Rightarrow 180^{\circ }n-2\times 180^{\circ }=5\times 90^{\circ }+195^{\circ }n-5\times 195^{\circ }$$
$$\Rightarrow 180^{\circ }n-360^{\circ }=450^{\circ }+195^{\circ }n-975^{\circ }$$
$$\Rightarrow 180^{\circ }n-360^{\circ }=450^{\circ }-975^{\circ }+195^{\circ }n$$
$$\Rightarrow 180^{\circ }n-360^{\circ }=-525^{\circ }+195^{\circ }n$$
$$\Rightarrow -525^{\circ }+195^{\circ }n=180^{\circ }n-360^{\circ }$$
$$\Rightarrow 195^{\circ }n=525^{\circ }+180^{\circ }n-360^{\circ }$$
$$\Rightarrow 195^{\circ }n-180^{\circ }n=525^{\circ }-360^{\circ }$$
$$\Rightarrow 15^{\circ }n=165^{\circ }$$
$$\Rightarrow n=\dfrac{165^{\circ }}{15^{\circ }}$$
$$\Rightarrow n=11$$
Therefore the polygon has 11 sides.

Note: So for the solution you need to know the basics about a polygon, a polygon is a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain and this solid plane figure is called a polygon.
The segments of a polygonal circuit are called its edges or sides, and the points where two edges meet are the polygon's vertices. The interior of a solid polygon is sometimes called its body. An n-gon is a polygon with n sides; for example, a triangle is a 3-gon. And if you know about its number of sides then you can easily find the summation of all its interior angles by the formula $$\left( n-2\right) 180^{\circ }$$.