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Heptagon word comes from two terms “Hepta” and “Gonia”. Hepta means seven and gonia mean angles. Thus, the word heptagon indicates seven angles that are combined to form a seven-sided polygon. Any two-dimensional closed shape that is formed using only straight lines is referred to as a polygon in geometry. Polygons are named according to the number of sides and the types of angles they have. A polygon can also be categorized depending on the number of sides as a triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, and decagon. They are also classified as concave or convex depending on their angles.

Heptagon is a polygon with 7 sides and 7 angles. It has seven vertices and the lengths of its sides can be the same or different. It is a closed figure and a regular heptagon is one with seven equally sized sides. It is also known as a septagon, where ‘septa’ is a Latin word for seven. There are 14 diagonals in a heptagon.

A heptagon is easily distinguished by its shape. It has seven angles and seven vertices. The fourteen diagonals of a regular heptagon make a seven-pointed star as shown below:

**Regular Heptagon: **A heptagon is referred to as a regular heptagon if all of its sides and angles are equal. The standard heptagon has 128.57° or 5π/7 radians of angle on each side.

**Irregular Heptagon: **If either the sides or the angles of a heptagon are not equal in measure, then it is an irregular heptagon.

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A heptagon is a polygon that has seven sides. A heptagon has seven straight edges that might be the same length or different lengths. These sides come together but do not cross or intersect. A seven-sided closed figure is created when the heptagon’s sides converge at its vertices.

The sum total of the interior angles of a heptagon, which has seven interior angles, is 900°. A heptagon may have some acute or obtuse internal angles. However, It is true for both regular and irregular heptagons that the total of a heptagon’s outer angles is 360°.

**Heptagon Interior Angles: **A polygon’s interior angle is the angle formed by its two sides, measured inside the polygon. All of the inner angles of a regular polygon are equal to one another. A polygon’s interior angles are proportional to the number of sides. Heptagon has seven inner angles as a result.

The formula for calculating the sum of interior angles of a polygon is (n − 2) × 180, where ‘n’ is the number of sides the polygon has.

For heptagon, n=7, so the measure of the sum of its interior angles is (7 − 2) × 180 = 5 x 180 = 900°. Consequently, for a ‘regular’ heptagon, the measure of each interior angle =\(\frac{900}{7}\)=128.57°.

A central angle of a regular polygon is the angle between the line segments from the endpoints of a side of the pentagon to its center. The measure of the central angle of a regular heptagon is approximately 51.43°.

**Heptagon Exterior Angles: **Exterior angles of a polygon are angles formed by one of its sides and the extension of the side adjacent to it. The sum of all the exterior angles in a polygon is equal to 360°. Thus, the sum of exterior angles of a heptagon is also 360°.

The formula for calculating the size of an exterior angle of a regular polygon = 360 ÷ number of sides.

Thus, for pentagon the measure of the exterior angle = 360 ÷ number of sides = 360 ÷ 7 = 51.4285714286°.

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Heptagons can be categorized as follows according to measures of the angles:

**Convex Heptagon: **All of the interior angles in a convex heptagon are less than 180°. They might be heptagons that are regular or irregular. The convex heptagon’s vertices are all inclined toward the outside.

**Concave Heptagon: **Any polygon with at least one interior angle greater than 180° is said to be concave. It has to have a minimum of four sides. The concave polygon typically has an erratic shape. Thus, in a concave heptagon, at least one of its interior angles is greater than 180°. Concave heptagons are always irregular. In a concave heptagon, at least one vertex faces inward.

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Heptagons have the following characteristics.

- There are 7 vertices, 7 edges, and 7 sides in a heptagon.
- The interior angles of a heptagon add up to 900°.
- A heptagon’s exterior angles add up to 360°.
- The internal angle of a standard heptagon is approximately 128.57°.
- A typical heptagon’s central angle is measured at about 51.43°. A central angle of a regular polygon is an angle whose vertex is the centre and whose rays, or sides, contain the endpoints of a side of the regular polygon.
- In a heptagon, there are 14 diagonals.
- Regular heptagons are always convex heptagons.
- In a heptagon, there are five triangles.

The formulas for heptagons include the equations to find out the area and perimeter of the heptagon. Let’s see them one by one.

**Area of Heptagon Formula**

The area of a regular heptagon can be calculated easily by using a straightforward formula. The area of the irregular heptagon is not easy to find out. We need to divide the irregular heptagon into some smaller polygons whose areas we can easily find out. The final area of the heptagon is the sum total of the area of all the polygons.

The area of the heptagon is the area enclosed by the seven segments of heptagon. The formula for the area of any polygon is given by \(Area={1\over{2}}nsa\). Here, “n” is the number of sides, “s” is the length of the sides, and “a” is the apothem.

For heptagon the number of sides is n = 7. Hence, the formula becomes,

\(Area={7\over{2}}sa=3.5sa\).

There is another formula for the area of heptagon when only one side is given. It is given by:

\(Area={7\over{4}}s^2cot({180\over{7}})^o\).

**Perimeter of Heptagon Formula**

A perimeter is a closed path that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional length. Thus, the perimeter of a heptagon is the sum of the length of all its sides. In the case of a regular heptagon, the sides are the same. Hence, the perimeter is given by,

\(Perimeter=7s\) where, \(s\) is the side length.

Now let’s see some examples of heptagon which you can practice.

**Example 1:** Find the area and perimeter of a heptagon having a side length of 33 m and the perpendicular distance from the centre to one of its sides as 28.5 m

**Solution: **The perpendicular distance from the centre to one of its sides is the apothem of the heptagon.

Therefore, a = 28.5 m and

s = 33 m.

\(Area=3.5{\times}28.5{\times}33=3291.75m^2\).

Perimeter is given by:

\(Perimeter=7{\times}33=231m\).

**Example 2:** Find the value of the missing angles in the following heptagon.

**Solution: **This is an irregular heptagon having different measures of angle. However, the property of interior angles of a heptagon states that the sum total of all the interior angles of a heptagon is equal to 900°.

Let the unknown angle be x.

Therefore, according to the property we can write,

100 + 140 + 130 + 135 + 115 + 150 + x = 900

770 + x = 900

x = 900 -770

x = 130°.

**Example 3:** Find the value of x, b and c.

This is an irregular heptagon having different measures of angle. However, the property of interior angles of a heptagon states that the sum total of all the interior angles of a heptagon is equal to 900°.

Therefore, according to the property we can write,

154.37 + 135.86 + 107.33 + 129.66 + 139.24 + 94.42 + x = 900

760.88 + x = 900

x = 900 – 760.88

x = 139.12°

Angles b and 94.42 are supplementary in nature.

Therefore, b + 94.42 = 180

b = 180 – 94.42 = 85.58°

Angles x and c are also supplementary in nature.

Therefore, x + c = 180

c = 180 – x

c = 180 – 139.12

c = 40.88°

Hope this article on the Heptagon was informative. Get some practice of the same on our free Testbook App. Download Now!

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Heptagon is a polygon with 7 sides and 7 angles. It has seven vertices and the lengths of its sides can be the same or different. It is a closed figure and a regular heptagon is one with seven equally sized sides.

Heptagon has seven inner angles as a result. The sum total of all the interior angles of a heptagon is equal to 900°. For regular heptagon, the measure of the interior angle is about 128.57°.

The formula for calculating the sum of interior angles is (n − 2) × 180, where is the number of sides. For regular heptagon, the measure of the interior angle is (7 − 2) × 180 = 5 x 180 = 900°. In heptagon, the sum of the interior angles is equal to 900°.

A seven sided polygon is called as heptagon.

In a concave heptagon, at least one of its interior angles is greater than 180°. They might be heptagons that are regular or irregular. In a concave heptagon, at least one vertex faces inward. All of the interior angles in a convex heptagon are less than 180°. They might be heptagons that are regular or irregular. The convex heptagon's vertices are all inclined toward the outside.

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