A diagonal is a line segment joining the two vertices of a polygon present on the opposite sides, and a diagonal is not a side of the polygon. Polygon is a two-dimensional (2D) flat figure which has all its sides closed and straight.
Every flat shape having sides and corners is a two-dimensional shape (2D shape); a circle is not a polygon. Triangle is a polygon having the least number of sides, which is 3. There are no fixed numbers of sides to be present, there can be numerous sides, but the figure formed by the sides must be closed irrespective of the length of sides provided to the figure.
So, let us understand the diagonals of a polygons.
In the given figure AB, BC, CD, DE, EA are sides while
AC, AD, BE, BD, and CE are diagonals.
Different Types Of Polygons
- Regular and irregular polygon: Regular polygons are those polygons with equal sides and angles; irregular polygon is a polygon whose sides and angles are not equal or the same.
- Convex and concave polygon: convex polygon is that polygon whose angle created by the sides from inside is less than 180 degrees; concave polygon are those whose angles are formed by its sides are more than 180 degrees.
- Equivalent polygon: equivalent polygons are those whose all sides are equal
- Equiangular polygon: Equiangular polygons are those whose all angles are equal
Expect from the above-given polygon types; there are many more based on the number of sides a polygon has.
Formula To Calculate No. Diagonals Of Polygon
Number of diagonals= n(n-3)/2
{n are the numbers of side}
There can be a polygon with n numbers of sides.
Derivation
Any vertex joined by two other vertices, so those connections cannot count as diagonals that’s why (n-3)
The total no. of n vertices gives us n(n-3) diagonals; then, we have to divide the final formula by 2 because every diagonal has two ends which will count the same diagonal twice.
Length Of Diagonals In Square
Pythagoras theorem is used to find the length of the diagonal of a square. A diagonal divides the square into two right-angle triangles; thus, the diagonal act as the hypotenuse.
Length of diagonal = √2x units = d
x is the length of the side.
Length Of Diagonals In Rectangle
The diagonal in the rectangle acts as a hypotenuse. It divides the rectangle into two right-angle triangles, so the Pythagoras theorem is used to find the diagonal length.
Formula to find length of diagonal of rectangle is (d) = √(l2 + b2)
Where d means the length of diagonal
I mean the length of the rectangle
b means base of the rectangle
Solved Examples
1. Example: If a polygon has 18 sideshow many diagonals it has?
Solution: number of sides = 18 (given in question) = n
Number of diagonals =n(n-3)/2
= 18(18-3)/2
= 18 *15/2
= 135 diagonals
2. Example: find the number of diagonals octagon can have?
Solution- octagon has 8 sides
so, number of sides (n) = 8
number of diagonals = n(n-3)/2
= 8(8-3)/2
= 40/2 = 20 diagonals
3. Example find the length of diagonal of square whose one side is 8 cm?
solution: since all sides of the square has equal length, we will take X = 8
length of diagonal of square = √2x
√2(8) =√16= 4 cm
Let’s Revise Quick Facts-
1: Does a triangle have diagonals?
Answer: a triangle has zero diagonals. A triangle does not have any non-adjacent vertices. No diagonal can be formed.
2: Why the circle is not a polygon?
Answer: circle being a two-dimensional shape, is not a polygon because it does not have any straight sides.
3: Is diagonal something different, and what shape it has?
Answer: diagonal is just a line segment formed by joining the two vertices of a. polygon
4: Can we use the same length of diagonal formula for square and rectangle?
Answer: No, since both square and rectangle are different polygons, we can’t use the same formula
Length of diagonal of the square: √2x
Length of diagonal of rectangle: √ (l2 + b2)
Conclusion
Therefore, we hope now you have a clear understanding of polygons diagonals. Practice questions regarding have this topic to ace all questions.
