What Is a Heptagon Area Formula Perimeter Formula and Solved Examples
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 Shape Definition Properties and Formulas
1. What is a heptagon?
A heptagon is a polygon with 7 sides, 7 vertices, and 7 interior angles. It is a type of closed 2D geometric figure. Heptagons can be regular (all sides and angles equal) or irregular (sides and angles not equal). The term comes from the Greek words “hepta” meaning seven and “gonia” meaning angles.
2. What is the sum of the interior angles of a heptagon?
The sum of the interior angles of a heptagon is 900°. It is calculated using the polygon formula (n − 2) × 180°, where n is the number of sides. For a heptagon, n = 7:
- (7 − 2) × 180°
- 5 × 180° = 900°
3. What is the measure of each interior angle of a regular heptagon?
Each interior angle of a regular heptagon measures approximately 128.57°. Since the sum of interior angles is 900°, divide by 7 sides:
- 900° ÷ 7 = 128.57° (approx.)
4. What is the formula for the area of a regular heptagon?
The area of a regular heptagon is given by A = (7/4)s² cot(π/7), where s is the side length. An approximate form is:
- A ≈ 3.634s²
- A ≈ 3.634 × 16 = 58.14 square units (approx.)
5. How many diagonals does a heptagon have?
A heptagon has 14 diagonals. The number of diagonals in any polygon is found using the formula n(n − 3)/2. For n = 7:
- 7(7 − 3)/2
- 7 × 4 ÷ 2 = 14
6. What is the difference between a regular and irregular heptagon?
A regular heptagon has all sides and angles equal, while an irregular heptagon does not. Key differences include:
- Regular: Each interior angle = 128.57°
- Irregular: Angles and sides vary
- Regular: Symmetrical shape
- Irregular: No uniform symmetry
7. What is the sum of the exterior angles of a heptagon?
The sum of the exterior angles of any heptagon is 360°. This rule applies to all polygons, regardless of the number of sides. In a regular heptagon, each exterior angle equals:
- 360° ÷ 7 = 51.43° (approx.)
8. How do you draw a regular heptagon?
A regular heptagon can be drawn by dividing a circle into seven equal central angles of 51.43° each. Steps include:
- Draw a circle using a compass.
- Mark the center and measure central angles of 51.43°.
- Mark seven equally spaced points on the circle.
- Join consecutive points with straight lines.
9. What are some real-life examples of a heptagon?
A common real-life example of a heptagon is the British 50 pence coin, which has seven sides. Other examples include:
- Certain architectural floor designs
- Decorative tiles and patterns
- Geometric logos
10. Is a heptagon the same as a septagon?
Yes, a heptagon and a septagon both refer to a 7-sided polygon. The term “heptagon” comes from Greek roots, while “septagon” comes from Latin. In mathematics, “heptagon” is the more commonly used term, but both names describe the same geometric shape.
