Heptagon: Definition, Classification, Polygon, Solved Examples

Heptagon is a polygon having seven sides and seven vertices. It is a closed figure that is also sometimes known as a septagon. Any closed figure that has at least 3 sides and 3 interior angles is known as a polygon. It is a curve that is closed and simple, and is made up only out of line segments.

Key Terms: Polygon, sides, vertices, angles, heptagon, interior, exterior, sum, area

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What is Polygon?

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The word ‘polygon’ comes from the Greek word polús, meaning ‘many’, and g?nía, meaning ‘angle’ or ‘corner’. A polygon is a two-dimensional figure that has a finite number of sides and vertices. It is a plane figure, and it is a closed curve. This means it has no gaps or breaks, and is a continuous figure, enclosed from all of its sides. There are many different types of polygons, discussed below.

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Classification of Polygons

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Generally, for any polygon having ‘n’ sides, has ‘n’ interior angles as well. Further, the interior angles and exterior angles as well as their sum can be found by the following formulas –

• Sum of its exterior angles = \(360°\)
• Each interior angle = \((n - 2) \times 180°/n\)
• Sum of its interior angles = \((n - 2) \times 180°\)

Polygons are further classified into triangles, quadrilaterals, pentagons, hexagons, heptagons, octagons, nonagons and decagons. There are also other polygons with more than 10 sides, such as a dodecagon, which has 12 sides. This classification is based on the number of sides that they have.

1) Triangles – 3 sides, vertices and interior angles.

There are 3 kinds of triangles as well; scalene triangles have all different sides and angles, an isosceles triangle has two equal sides and angles with the third one being a different length, and an equilateral triangle has sides of equal length, its inside angles making a sum of 60°.

2) Quadrilateral – 4 sides and interior angles.

In a quadrilateral, the sum of all its internal angles equals 360°. There are many shapes that are quadrilaterals, such as rectangles, squares, trapezium, and parallelograms.

3) Pentagon – 5 sides and interior angles.

The sum of interior angles in the case of a pentagon is always 540°.

4) Hexagon – 6 sides and interior angles.

In a hexagon, all of its 6 sides have to be equal, which means their interior angles are equal as well. The sum of interior angles in a hexagon is always 720°.

5) Heptagon – 7 sides and interior angles.

The sum of interior angles in a heptagon is always 900°.

6) Octagon – 8 sides and 8 interior angles.

The sum of interior angles in a octagon is always 1080°.

7) Nonagon – 9 sides and interior angles.

The sum of interior angles in a nonagon is always 1260°.

8) Decagon – 10 sides and interior angles.

The sum of interior angles in a decagon is always 1440°.

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Heptagons

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The word heptagon is derived from the word ‘Hepta’, meaning seven, and ‘Gonia’ meaning angles. A polygon that has 7 sides, 7 vertices and 7 interior angles is known as a heptagon or septagon. It also has 14 diagonals.

If ‘a’ is the length of the side of a regular heptagon, then,

Area of a heptagon = \(\frac{7}{4} a^2 \cot \frac{\pi}{7}\) sq units, or \(3.634\ a^2\) square units

Perimeter of a heptagon = \(7a\) units

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Types of Heptagons

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Heptagons can be of many types, based on the properties of their angles and sides.

Based on the location of their diagonals, they can be convex or concave. A heptagon is convex when all its diagonals exist inside the polygon itself. None of its internal angles would be more than 180°. It is concave when some of its diagonals lie outside the polygon, and one of its interior angles would be more than 180°.

Convex and Concave Heptagons

Based on the equalness of its sides and angles, a heptagon can be regular or irregular. A heptagon is regular when all its sides and angles are equal. Each of its interior angles would equal 128.571° and each exterior angle would be 51.428°. It is irregular when its angles and sides are not equal.

Regular and Irregular Heptagons

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Things to Remember

• A polygon is a closed curve with a finite number of sides. Polygons can be triangles, quadrilaterals, pentagons, hexagons, heptagons, octagons, nonagons and decagons.
• A heptagon has 7 sides, 7 vertices and 7 interior angles.
• Based on the properties of its angles and sides, a heptagon can be regular or irregular, convex or concave.
• Area of a heptagon is \(3.634\ a^2\) sq units.
• Perimeter of a heptagon is 7a units.

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