How to Find the Area and Perimeter of a Nonagon with Examples
The concept of nonagon plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding a nonagon helps students strengthen their foundation in polygons, geometry calculations, and logical reasoning questions for classes 6–10 and even competitive exams like JEE.
What Is a Nonagon?
A nonagon is defined as a polygon with nine straight sides and nine angles. The word "nonagon" comes from Latin ("nonus" for nine) and Greek ("gon" for angle/sides). You’ll find this concept applied in geometry, architectural design, and even in puzzle problems where recognizing shapes quickly leads to faster solutions.
Properties of a Nonagon
| Property | Regular Nonagon | Irregular Nonagon |
|---|---|---|
| Number of sides | 9 | 9 |
| All sides equal? | Yes | No |
| All angles equal? | Yes (140° each) | No |
| Sum of interior angles | 1260° | 1260° |
| Number of diagonals | 27 | 27 |
Key Formula for Nonagon
Here’s the standard formula for the area of a regular nonagon (all sides equal):
Area = \( \frac{9}{4}a^2 \cot\left(\frac{\pi}{9}\right) \)
where ‘a’ is the length of one side.
Perimeter = 9 × a
Sum of interior angles = (9 − 2) × 180° = 1260°
Each interior angle (regular) = 140°
Each exterior angle (regular) = 40°
How to Calculate Area and Perimeter: Step-by-Step
- Identify if the nonagon is regular (all sides equal).
If yes, use the formula directly. If not, break the shape into triangles or use coordinates. - For a regular nonagon with side ‘a’ (example: a = 6 cm):
Area = \( \frac{9}{4} \times (6^2) \times \cot\left(\frac{\pi}{9}\right) \)Area ≈ 2.25 × 36 × 2.747 = 222.63 cm² (using \( \cot\left(\frac{\pi}{9}\right) \approx 2.747 \)) - Perimeter = 9 × side = 9 × 6 = 54 cm
Types of Nonagons
Nonagons can be:
- Regular nonagon: All sides and angles are equal.
- Irregular nonagon: Sides/angles are NOT all equal, but still has 9 sides and 9 angles.
- Convex nonagon: All angles less than 180°, vertices "point outwards".
- Concave nonagon: At least one angle greater than 180°, looks "dented" inward.
How to Draw a Nonagon
- Use a compass to draw a circle (this will be the circumcircle).
- Mark a starting point on the circle as vertex 1.
- With a protractor, measure 40° arcs from the center (as each central angle is \( 360°/9 = 40° \)). Mark these on the circle.
- Join consecutive points – you get a regular nonagon!
Everyday Examples of Nonagons
- US Steel Tower in Pittsburgh is shaped like an irregular nonagon.
- Certain coins, window designs, and wall decorations use the nonagon shape.
- Puzzles and creative art use both regular and irregular nonagons as patterns.
Speed Trick or Memory Aid
To remember the area formula for a regular nonagon, just recall: “9 over 4, side squared, cot pi over 9.” Use it as a rhyme or jot in your formula book. For quick MCQ calculation, round \( \cot(\pi/9) \) to 2.747, and multiply.
Tip: Sum of angles in any polygon = (n − 2) × 180°. For a nonagon, just plug in n = 9.
Try These Yourself
- Calculate the area of a regular nonagon with each side 10 cm.
- Draw a nonagon using a compass and protractor. Mark all nine vertices.
- Is it possible to have a regular concave nonagon?
- Find the sum of the exterior angles of a nonagon (regular or irregular).
Frequent Errors and Misunderstandings
- Thinking "irregular" nonagons don’t have constant angle sum—always 1260° for any nonagon.
- Mixing up diagonals: Use \( n(n-3)/2 \); for n = 9, result is 27.
- Assuming nonagon can tessellate a plane alone—it can’t!
Relation to Other Concepts
The idea of nonagon connects closely with topics such as Types of Polygons, Interior Angles of a Polygon, and Area of Polygon. Mastering this helps with geometric constructions, tessellations, and advanced proofs in higher classes and competitive exams.
Classroom Tip
Easy mnemonic: "Nona means 9." On the board, always draw sides with tick marks to keep count, especially while learning polygons up to dodecagon. Vedantu’s teachers use “polygon flashcards” to reinforce shape names, angles, and formulas in fun quizzes.
We explored nonagon—from definition, formula, examples, mistakes, and connections to other important geometry topics. Keep practicing problems, try drawing nonagons with different side lengths, and use Vedantu’s online resources to boost your exam skills and overall confidence with polygons!
FAQs on Nonagon: Definition, Properties, Formula, & Examples
1. What is a nonagon in Maths?
A nonagon, also known as an enneagon or 9-gon, is a polygon with nine sides and nine angles. It can be regular (all sides and angles equal) or irregular (sides and angles of varying measures).
2. How many sides and angles does a nonagon have?
A nonagon has nine sides and nine angles. The sum of its interior angles is always 1260 degrees.
3. What is the formula for the area of a regular nonagon?
The area (A) of a regular nonagon with side length 'a' is calculated using the formula: A = (9/4) × a² × cot(π/9). Alternatively, using the inradius 'r', the area is A = (9/2)ar, and using the circumradius 'R', the area is A = (9/2)R²sin(2π/9).
4. What is the sum of the interior angles of a nonagon?
The sum of the interior angles of any nonagon (regular or irregular) is always 1260 degrees. This can be derived from the general formula for the sum of interior angles of an n-sided polygon: (n-2) × 180°.
5. How is a regular nonagon different from an irregular nonagon?
In a regular nonagon, all nine sides are of equal length, and all nine interior angles are equal (140° each). In an irregular nonagon, the sides and angles have different measures, although the sum of interior angles remains 1260°.
6. How many diagonals does a nonagon have?
A nonagon has 27 diagonals. This can be calculated using the formula: n(n-3)/2, where 'n' is the number of sides (in this case, 9).
7. What is the perimeter of a regular nonagon?
The perimeter (P) of a regular nonagon with side length 'a' is simply 9a. This is because all sides are of equal length in a regular nonagon.
8. What are some real-world examples of nonagons?
While perfectly regular nonagons are rare, irregular nonagonal shapes can be found in some architectural designs and naturally occurring patterns. Precise examples require further research and may not always be readily apparent.
9. How do I calculate the measure of each interior angle in a regular nonagon?
For a regular nonagon, each interior angle measures 140°. This can be found using the formula: [(n - 2) × 180°] / n, where 'n' is the number of sides (9).
10. What is the difference between a convex and a concave nonagon?
A convex nonagon has all its interior angles less than 180°. A concave nonagon has at least one interior angle greater than 180° (a reflex angle). All regular nonagons are convex.
11. What are the steps involved in constructing a regular nonagon using a compass and straightedge?
Precise construction of a regular nonagon using only a compass and straightedge is not possible. Approximation methods exist, but they involve iterative processes and are beyond the scope of a simple answer. More advanced geometric tools or techniques would be needed for an accurate construction.
12. How is the area of an irregular nonagon calculated?
Calculating the area of an irregular nonagon requires breaking it down into smaller, simpler shapes (like triangles) whose areas are easier to determine. The sum of the areas of these smaller shapes will give the total area of the irregular nonagon. More complex methods may involve using calculus or numerical integration techniques.
