How Many Angles and Sides Does a Heptagon Have?
The concept of heptagon plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding heptagons helps students distinguish between different polygon types and solve a wide range of geometry problems, from angle calculations to perimeter and area.
What Is a Heptagon?
A heptagon is defined as a polygon with seven straight sides and seven angles. Sometimes, it is also called a "septagon." Heptagons can be seen in mathematics, science, engineering, and even in daily life objects. The key idea is that each side must be straight, and all sides join together to form a closed figure. You’ll find this concept applied in geometry, polygon classification, and measurement topics.
Heptagon Properties
| Property | Details |
|---|---|
| Number of Sides | 7 |
| Number of Angles | 7 |
| Sum of Interior Angles | 900° |
| Sum of Exterior Angles | 360° |
| Diagonals | 14 |
| Types | Regular, Irregular, Convex, Concave |
| Alternate Name | Septagon |
Key Formula for Heptagons
Here are important formulas for a heptagon:
- Sum of interior angles: (7 − 2) × 180° = 900°
- Each angle in regular heptagon: 900° ÷ 7 ≈ 128.57°
- Number of diagonals: n(n−3)/2 = 7×4/2 = 14
- Perimeter (regular): 7 × a (where a = side length)
- Area (regular): \(\frac{7}{4}a^2 \cot(\pi/7)\) or approximately 3.634 × a²
Types of Heptagons: Regular vs Irregular
| Type | Sides & Angles | Features |
|---|---|---|
| Regular Heptagon | All equal | Equal sides & angles, each angle ≈ 128.57° |
| Irregular Heptagon | Unequal | Sides & angles are not equal |
| Convex Heptagon | All angles < 180° | All diagonals inside |
| Concave Heptagon | At least 1 angle > 180° | Some diagonals outside |
How to Draw a Heptagon (Step-by-Step)
- Draw a large circle as the base guide.
- Mark a center point in the circle.
- Use a protractor to mark 51.43° angles at the center (since 360°/7 ≈ 51.43°) to get 7 equal arcs.
- Join each arc point with straight lines.
- Erase the circle and admire your heptagon!
Example Problem: Find Perimeter and Area
Let’s try a quick example: Find the perimeter and area of a regular heptagon with each side = 5 cm.
1. Perimeter = 7 × 5 = 35 cm2. Area ≈ 3.634 × (5)² = 3.634 × 25 = 90.85 cm²
Speed Trick or Vedic Shortcut
Angle check shortcut: To quickly find the sum of interior angles in any polygon, multiply (number of sides – 2) by 180°. For heptagon: (7–2) × 180° = 900°. This saves time in MCQs and geometric proofs.
Try These Yourself
- How many diagonals does a heptagon have?
- Find the measure of each interior angle in a regular heptagon.
- Draw a convex and a concave heptagon—can you spot the difference?
- Calculate the perimeter of a regular heptagon if each side is 8 cm.
Frequent Errors and Misunderstandings
- Confusing a heptagon with hexagon (6 sides) or octagon (8 sides).
- Using 7 instead of (7–2) in the angle sum formula.
- Forgetting that not all heptagons have equal sides (irregular type).
Heptagons in Real Life
You can spot heptagons in coins, signs, tile designs, and even in nature (like some flowers or crystals). Knowing what a heptagon looks like helps in quick identification and makes geometry more interesting. Vedantu often uses real-world visuals to help make concepts clear!
Relation to Other Geometry Topics
Understanding the heptagon is useful when learning about Types of Polygons, Interior Angles of a Polygon, and Regular Polygon calculations. These skills are connected to many exam questions and future geometry chapters.
Quick Classroom Tip
To remember how many sides a heptagon has, think “Hepta = 7.” Draw and color polygons with different numbers of sides for quick practice. Vedantu’s teachers use such memory cues to help you master tricky names!
We explored heptagon—from definition, formulas, step-by-step drawing, and common mistakes. Practicing with Vedantu classes and free question banks will help you become confident at identifying, drawing, and solving problems with heptagons and other polygons.
Related Pages to Explore
FAQs on Heptagon – Definition, Properties, Formula, and Examples
1. What is a heptagon?
A heptagon is a polygon with seven sides and seven angles. It's sometimes called a septagon. The sum of its interior angles is always 900 degrees. Heptagons can be regular (all sides and angles equal) or irregular (sides and angles of varying lengths and measures).
2. What are the properties of a heptagon?
Key properties of a heptagon include:
- Seven sides
- Seven angles
- Sum of interior angles: 900 degrees
- Sum of exterior angles: 360 degrees
- Number of diagonals: 14
3. How do I calculate the sum of interior angles in a heptagon?
The sum of interior angles of any polygon with 'n' sides is given by the formula: (n-2) × 180°. For a heptagon (n=7), the sum is (7-2) × 180° = 900°.
4. What is the measure of each interior angle in a regular heptagon?
In a regular heptagon, all interior angles are equal. To find the measure of each angle, divide the total sum of interior angles (900°) by the number of angles (7): 900° / 7 ≈ 128.57°.
5. How many diagonals does a heptagon have?
A heptagon has 14 diagonals. The formula to calculate the number of diagonals in a polygon with 'n' sides is: n(n-3)/2. For a heptagon (n=7), this is 7(7-3)/2 = 14.
6. What is the difference between a regular and an irregular heptagon?
A regular heptagon has all seven sides of equal length and all seven angles of equal measure. An irregular heptagon has sides and angles of varying lengths and measures.
7. How do I calculate the perimeter of a regular heptagon?
The perimeter of any polygon is the sum of the lengths of its sides. For a regular heptagon, where all sides have the same length ('a'), the perimeter is simply 7a.
8. How do I calculate the area of a regular heptagon?
The area of a regular heptagon with side length 'a' can be calculated using the formula: Area ≈ 3.634a². More precise calculations require using the apothem (the distance from the center to the midpoint of a side).
9. What is a convex heptagon?
A convex heptagon is a heptagon where all interior angles are less than 180°. All diagonals lie within the polygon.
10. What is a concave heptagon?
A concave heptagon has at least one interior angle greater than 180°. At least one diagonal lies outside the polygon.
11. Are heptagons commonly found in real-world objects?
While less common than other polygons like triangles, squares, or hexagons, heptagons can be found in some real-world objects, although often as approximations. Some examples include certain types of man-made structures and designs.
12. Can a regular heptagon be constructed using only a compass and straightedge?
No, a regular heptagon cannot be constructed using only a compass and straightedge. It requires more advanced techniques or approximations.
