What is a Polygon?
A polygon is a closed 2D shape formed by line segments. The term ‘polygon’ is made of two terms ‘poly’ meaning many and ‘gon’ meaning angles. Thus, a polygon contains many angles.
Polygons are classified based upon the number of sides each shape has. The simplest polygons include a triangle (3 sides), a quadrilateral (4 sides), a pentagon (5 sides) and so on.
What is a Heptagon?
A heptagon is a polygon with seven sides. Sometimes, a heptagon is also called a ‘septagon’. A heptagon has seven edges or sides, seven angles and seven vertices. When all seven sides of a heptagon are equal in length then it is called a regular heptagon. All seven angles of a regular heptagon are also equal in measure.
Angles in a Heptagon
There are two types of angles in a polygon, interior angles and exterior angles. There are seven interior angles, and correspondingly seven exterior angles in heptagons.
We can also observe that the sum of all interior angles of a heptagon is 900° and just like any other polygon the sum of all the exterior angles of a heptagon is 360°.
Classification of Heptagons
A heptagon is classified on the basis of side length and angle measure.
On the basis of side length, there are two types of heptagons:
- Regular Heptagon: In a regular heptagon all sides and angles are congruent.
- Irregular Heptagon: When any two sides of a heptagon differ in length, then, it is called an irregular heptagon.
On the basis of angle measurements, there are two types of heptagons:
- Convex Heptagon: In a convex heptagon, each interior angle is less than 180°.
- Concave Heptagon: If any of the interior angles exceed 180°, then it is a concave heptagon.
Perimeter of a Heptagon
The perimeter of a geometric shape is defined as the total length of the boundary of that shape. Here the heptagon has seven sides along its boundary so the sum of length of all the sides of a heptagon is its perimeter.
Perimeter of heptagon, P = sum of the length of all sides
We know that all the sides of a regular heptagon are equal,
Hence, the perimeter of a regular heptagon, P = 7 \( \times\) side length
Rapid Recall
- Heptagons have seven sides, seven angles and seven vertices.
- In a regular heptagon, all the sides and angles are congruent.
- The sum of all interior angles is 900°, and each angle of the regular heptagon is 128.57 in degrees and \( \frac{5\pi}{7}\) in radians.
- The sum of all exterior angles of a heptagon is 360° and each exterior angle of a regular heptagon is 51.43 in degrees and \( \frac{2\pi}{7}\) in radians.
- Perimeter of regular heptagon (P) = 7 \( \times\) side length.
Solved Heptagon Examples
Example 1: If the perimeter of a regular heptagon is 56 inches, find the side length.
Solution:
P = 7 \( \times\) side length Write the formula for perimeter
56 = 7 \( \times\) side length Substitute 56 for P
\(\frac{56}{7}\) = \(\frac{7~\times~side~length}{7}\) Divide each side by 7
8 = side length Simplify
So, the side length of the regular heptagon is 8 inches.
Example 2: If the length of one side of a regular heptagon is 12 feet, find its perimeter.
Solution:
P = 7 \(\times\) side length Write the formula for perimeter
P = 7 \(\times\) 12 Substitute 12 for side length
P = 84 Multiply
So, the perimeter of the regular heptagon is 84 feet.
Example 3: Find the perimeter of the heptagon coin of side length 8 mm.
Solution:
The coin is in the shape of a regular heptagon, to find the perimeter of the coin use the perimeter of the regular heptagon formula.
P = 7 \(\times\) side length Write the formula for perimeter
P = 7 \(\times\) 8 Substitute 8 for side length
P = 56 Multiply
So, the perimeter of the coin is 56 millimeters.
Example 4: Two interior angles of a heptagon are 120° and 200°. Can you name the type of heptagon?
Solution:
Here one angle is more than 180°, that is, a reflex angle. Hence the heptagon is concave heptagon.
The measure of each interior angle of a regular heptagon is 128.57°.
There are seven vertices in a heptagon.
There are seven interior angles in a heptagon.
Equilateral triangles are the only regular polygons with three sides.
On the basis of side length: regular and irregular heptagons
On the basis of angle measure: convex and concave heptagons