Geometry - Nonagon
- What is Nonagon?
- What is a Regular Nonagon
- What is an Irregular Nonagon
- Interior Angles and Exterior Angles of a Regular Nonagon
- Solved Examples
- Practice Problems
- Frequently Asked Questions
What is Nonagon?
A polygon with `9` sides and `9` angles is called a nonagon. The two words Nona and Gon that make up the word Nonagon stand for nine and sides, respectively.
The figure above is a nonagon `ABCDEFGHI` having sides `AB,`` BC,`` CD,`` DE,`` EF,`` FG,`` GH,`` HI,` and `IA`. The angles in the given nonagon are `∠A,`` ∠B,`` ∠C,`` ∠D,`` ∠E,`` ∠F,`` ∠G,`` ∠H,` and `∠I`. Lines `AC` and `BD` are two of the diagonals of the nonagon.
A nonagon has `27` diagonals as shown above.
What is a Regular Nonagon
Any nonagon with equal sides and equal-sized angles is referred to as a regular nonagon. Below is a diagram of a regular nonagon:
What is an Irregular Nonagon
An irregular nonagon is a type of nonagon having unequal sides and unequal measures of angles. Given below are some examples of irregular nonagons.
Interior Angles and Exterior Angles of a Regular Nonagon
For a regular nonagon, each of its interior angles measures `140°`. Therefore, the sum of all interior angles of a regular nonagon is `1260°`.
For a regular nonagon, each of its exterior angles measures `40°`. Therefore, the sum of all exterior angles of a regular nonagon is `360°`.
Solved Examples
Example `1`: Check whether the given figure is a nonagon or not. Give reasons.
Solution:
The given figure is a nonagon as it is a closed figure with `9` sides.
Example `2`: Calculate the length of the side of the regular nonagon, if the perimeter of the nonagon is `36` cm.
Solution:
In a regular nonagon, all the sides are of equal length.
Perimeter of a figure `=` Sum of all the sides
Perimeter of nonagon `= 36` cm.
Therefore, the measure of each side of the nonagon `= 36 ÷ 9 = 4` cm.
Practice Problems
Q`1`. How many sides does a nonagon have?
- Six
- Eight
- Nine
- Ten
Answer: c
Q`2`. What is the sum of the interior angles in a nonagon?
- `1400^\circ`
- `1440^\circ`
- `1260^\circ`
- `1080^\circ`
Answer: b
Q`3`. If one interior angle of a regular nonagon is `140^\circ`, what is the measure of each exterior angle?
- `20^\circ`
- `30^\circ`
- `40^\circ`
- `50^\circ`
Answer: c
Q`4`. What is the degree measure of each interior angle in a regular nonagon?
- `120^\circ`
- `140^\circ`
- `150^\circ`
- `160^\circ`
Answer: a
Q`5`. If the perimeter of a regular nonagon is `45` cm, what is the length of each side?
- `4.5` cm
- `5` cm
- `6` cm
- `7.5` cm
Answer: b
Frequently Asked Questions
Q`1`. How many sides does a nonagon have?
Answer: A nonagon have `9` sides.
Q`2`. What is the sum of the interior angles of a nonagon?
Answer: The sum of interior angles in a polygon with `n` sides is given by the formula \( (n-2) \times 180 \) degrees. For a nonagon `n=9`, therefore the sum of the interior angles is \( (9-2) \times 180 = 1260 \) degrees.
Q`3`. What is the degree measure of each interior angle in a regular nonagon?
Answer: The formula for the measure of each interior angle in a regular polygon is \( \frac{(n-2) \times 180}{n} \), where `n` is the number of sides. For nonagon `n=9`.
\(\begin{align*}
\text{Each angle}& =\frac{(9-2) \times 180}{9} \\& = \frac{7 \times 180}{9} \\
& = \frac{1260}{9} \\
& = 140^\circ
\end{align*}\)
Q`4`. What is the sum of the exterior angles of a nonagon?
Answer: The sum of all exterior angles in a polygon is always equal to `360^\circ`. Therefore, the sum of the exterior angles of a nonagon is `360^\circ`.