Question

Find the number of diagonals of a decagon?

Answer 1

Hint: Here in this question, we have to tell how many diagonals are there in the decagon shape. diagonal is an abstract object or line that connects two points or vertices of a polygon except sides of a polygon. This can be solved, by using a diagonal formula i.e., \[\dfrac{{n\left( {n - 3} \right)}}{2}\] where \[n\] is the number of vertices. on substituting the \[n\] values and by further simplification we get the required solution.

Complete step-by-step answer:
A decagon is a ten-sided polygon with ten vertices and ten angles.
A diagonal is a straight line connecting the opposite corners of a polygon through its vertex.
 Otherwise, diagonal is a line segment connecting two non-adjacent vertices of a polygon. It joins the vertices of a polygon excluding the edges of the figure.
Diagonals for polygons of all shapes and sizes can be made and for every shape; there is a formula to determine the number of diagonals.
Number of diagonals in a polygon with n vertices = \[\dfrac{{n\left( {n - 3} \right)}}{2}\].
In decagon, number of vertices \[n = 10\], then
\[ \Rightarrow \dfrac{{10\left( {10 - 3} \right)}}{2}\]
\[ \Rightarrow \dfrac{{10\left( 7 \right)}}{2}\]
\[ \Rightarrow \dfrac{{70}}{2}\]
On simplification, we get
\[ \Rightarrow 35\]
Hence, there are a total 35 diagonals in the decagon.
So, the correct answer is “35”.

Note: A diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal.
Diagonal polygons are also found by using a permutation and combination method.
In decagon, total vertices are 10
Now, the 10 vertices can be joined with each other by \[^{10}{C_2}\] ways.
Remember the formula of combination i.e., \[^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}\], then
\[ \Rightarrow {^{10}}{C_2} = \dfrac{{10!}}{{\left( {10 - 2} \right)!2!}}\]
\[ \Rightarrow {^{10}}{C_2} = \dfrac{{10 \times 9 \times 8!}}{{8!2!}}\]
\[ \Rightarrow {^{10}}{C_2} = \dfrac{{10 \times 9}}{2}\]
\[ \Rightarrow {^{10}}{C_2} = \dfrac{{90}}{2} = 45\]
There are 45 possible diagonals for a dodecagon which includes its sides.
So, the exact number of diagonals in decagon is:
\[ \Rightarrow 45 - 10\]
\[ \Rightarrow 35\]
There are 35 diagonals in the decagon shape.