Introduction to Geometry
Before we begin, let's understand shapes and basic geometry. Geometry is an extended field which can further be defined as a branch of mathematics that handles measurement, properties, relationship of points, lines, angles, surfaces etc of any given shape or term. It can help us know the characteristics of any given shape or a single line as well.
Geometry also helps us in studying the properties of given elements that remain invariant under specified transformations. Within the explanation of geometry we also include 2D shapes which include figures or flat shapes such as squares, circles, and triangles, being called a part of flat geometry, these shapes only have 2 dimensions which is length and width.
About Polygon
Now what is a polygon? A polygon is any closed shape with straight sides.Various examples of a polygon are rectangles, triangles, hexagons, and octagons. If we trace back to the meaning of the word “polygon” it can be noted that the word comes from the Greeks in the late 16th century via the word “polygon, neuter'' whose meaning was “many-angled”. Closed shapes cannot and do not have any gaps or openings and it can be said to be a free flowing incomplete figure which will be considered as a polygon when enclosed from all sides.
A heptagon is a closed figure with seven sides. A heptagon has seven sides and seven angles. All sides of a regular heptagon are equal.
Types of Polygons
In Geometry, we study different types of shapes. A polygon is any closed geometrical shape made up of straight lines that can be drawn on a flat surface, like a piece of paper. A circle or any other shape that includes a curve are not polygons.
Polygons are classified into different types depending on the number of sides they have:
(image will be updated soon)
Triangle (Polygon having 3 sides and 3 interior angles)
Three-Sided Polygons are called “triangles”.
A three-sided polygon is a triangle. Triangles also include different types, including:
Equilateral:
all the sides of the given polygon are of equal lengths, and the sum of all the internal angles are 60°.
Isosceles:
having two equal sides, with the third one a different length. Both of the internal angles are equal.
Scalene:
all of the three sides and internal angles, are different.
Triangles have internal angles which can also be defined upon those characteristics. Internal angles mean that the sides, whenever added up, will always be equal to or add up to 180° only. Any other deviation from this number will lead to the trainage being a characteristic of the other triangle. Further, a triangle with only acute internal angles is called an acute or acute-angled triangle. The one with one obtuse angle and two acute angles is called obtuse-angle and one with a right angle is known as right-angled.
Each of these will be either called equilateral, isosceles or scalene.
Quadrilateral (Polygon having 4 sides and 4 interior angles)
A quadrilateral has four sides and the family of quadrilaterals includes shapes such as square, rectangle, rhombus and other parallelograms, trapezium/trapezoid and kite.
The sum of all the internal angles of all quadrilaterals adds up to give 360°. For example, a square is a quadrilateral in which all slides are equal and all angles are equal to 90 degrees.
Pentagon (Polygon having 5 sides and 5 interior angles)
The Pentagon is a five-sided shaped Polygon and it can be obtained by joining each point of the five-line segments in the same plane. A regular pentagon has all five sides of the Polygon equal in length and if one side is not of the equal length then it cannot be qualified to be called a pentagon and then be referred to as an unequal pentagon. The sum of the interior angle of a pentagon is 540°.
Hexagon (Polygon having 6 sides and 6 interior angles)
A Hexagon is a Polygon that has 6 sides and 6 vertices, which is six sides and all has to be equal in length. The interior angles and exterior angles are also equal in measurement and the sum of the interior angle of a hexagon is 720°.
Heptagon (Polygon having 7 sides and 7 interior angles)
Octagon (Polygon having 8 sides and 8 interior angles)
Nonagon (Polygon having 9 sides and 9 interior angles)
Decagon (Polygon having 10 sides and 10 interior angles)
In General, A Polygon with ‘n’ Sides Has:
‘n’ interior angles.
Sum of interior angles = (n - 2) × 180°
Each interior angle of a regular polygon = \[\frac{(n-2) \times 180°}{n}\]
Sum of exterior angles = 360°
In this article, we will learn about the seven-sided polygon called “heptagon” with its proper definition, shape, number of sides, properties, its formula of perimeter and area in detail.
Definition of Heptagon
(image will be updated soon)
A heptagon is a polygon which has seven sides and seven angles. The word “heptagon” is made up of two words, namely ‘Hepta’ and ‘Gonia’, which means seven angles.
A heptagon can also be classified as regular and irregular types. If all the sides and including the interior angles of a heptagon are equal then it makes up a regular heptagon, it also consists of congruent sides and angles.
Given by the properties, heptagons can also be classified as either concave or convex. Convex heptagon is a polygon with seven sides which engulfs all of its diagonals inside the heptagon.
Irregular heptagon, on the other hand, is an irregular heptagon whose sides and angles are irregular and not congruent.
Sometimes the heptagon is also known as “septagon”.
Since, heptagon has 7 sides therefore,
Sum of interior angles = (n - 2) × 180°
= (7 - 2) × 180° = 5 × 180°
= 900°
Each interior angle of a regular heptagon = \[\frac{(n-2) \times 180°}{n}\]
= \[\frac{(7-2) \times 180°}{5}\]
= \[\frac{900}{5}\]
= 128.571°
Sum of exterior angles = 360°
Shapes of Heptagon
Depending on the sides, angles and vertices, heptagon shapes are classified as:
Regular heptagons
Irregular heptagons
Regular Heptagon
(image will be updated soon)
To be a regular heptagon the heptagon must have:
seven congruent sides (sides of equal length)
seven congruent interior angles (each measuring 128.571°)
seven congruent exterior angles of 51.428°
Note: Regular heptagons do not have parallel sides.
Irregular Heptagon
Irregular heptagons are the heptagon having different side lengths and angle measures.
(image will be updated soon)
Irregular heptagons can be a convex heptagon or a concave heptagon:
Convex Heptagon:
A heptagon having not any internal angles more than 180°.
Concave Heptagon:
A heptagon having one interior angle more than 180°.
(image will be updated soon)
Properties of Heptagon
It has seven sides, seven vertices and seven interior angles.
It has 14 diagonals.
The sum of all interior angles is 900°.
The sum of the exterior angles is 360°.
Regular heptagon has all seven sides of equal length.
Each interior angle of a regular heptagon measures 128.571°.
Irregular heptagons have different side lengths and angle measures.
All diagonals of the convex heptagon lie inside the heptagon.
some diagonals of concave heptagon may lie outside the heptagon.
The Perimeter of a Heptagon
The perimeter of a heptagon is the sum of lengths of its seven sides.
For a regular heptagon, since the length of all seven sides are equal.
Therefore, the perimeter of a regular heptagon = 7 × (side length) units.
Area of a Heptagon
Area of the heptagon is the region covered by the sides of the heptagon.
For a regular heptagon, its area can be calculated by:
(image will be updated soon)
1. If Measure of Side Length and Apothem are Given, Then:
(Apothem: a line from the centre of a regular polygon at right angles to any of its sides.)
Area of heptagon = \[\frac{7}{2}\] × (side length) × (apothem) units²
Area of heptagon = \[\frac{1}{2}\] × (perimeter of heptagon) × (apothem) units²
2. If Only Measure of Side Length is Given, Then:
Area of heptagon = \[\frac{7}{4}\] cot \[\frac{\pi}{7}\] × (side length)² units²
OR,
Area of heptagon = 3.634 × (side length) 2 units2
Solved Problems:
Q.1. Find the perimeter and area of a regular heptagon whose side is 7 cm?
Solution: Given, side of heptagon = 7 cm
Perimeter of a regular heptagon = 7 × (side length) units
= 7 × 7 cm
= 49 cm
And,
Area of heptagon = 3.634 × (side length)² units²
= 3.634 × (7)²
= 178.066 cm²
Hence, the perimeter and area of a regular heptagon whose side is 7 cm is 49 cm and 178.066 cm² respectively.
Conclusion
Solving the concept of Heptagon with examples will certainly help students in understanding the chapter, which has been done in the above content.
