| A square has 2 diagonals | An octagon has 20 diagonals |
A polygon's diagonals are line segments from one corner to another [but not the edges].
The number of diagonals of an n-sided polygon is:
n[n − 3] / 2
Examples:
- a square [or any quadrilateral] has 4[4−3]/2 = 4×1/2 = 2 diagonals
- an octagon has 8[8−3]/2 = 8×5/2 = 20 diagonals.
- a triangle has 3[3−3]/2 = 3×0/2 = 0 diagonals.
Try it Yourself:
| A diagonal can actually be outside the polygon, which happens with some concave polygons. |
Geometry Index
In geometry, a decagon is known as a ten-sided polygon or ten-gon. The sum of the interior angles of a simple decagon is 1440° and the sum of the exterior angles of a decagon is 360°.
Meaning of Decagon
A decagon is a ten-sided polygon with ten vertices and ten angles. Thus, a decagon shape can be defined as a polygon having ten sides, ten interior angles, and ten vertices. Based on the sides of a decagon, they are broadly classified into regular decagons and irregular decagons. A regular decagon has 35 diagonals and 8 triangles. The location of these diagonals and triangles is explained in the later sections of this article.
Types of Decagon
Decagons can be categorized as regular and irregular decagons based on side-length and angle measurements. There are three possible classifications of decagon that are given below:
- Regular and Irregular Decagons
- Convex and Concave Decagons
- Simple and Complex Decagons
Regular Decagon
A regular decagon is a polygon along with20 equal sides and 10 vertices. The sides and angles are congruent in a regular decagon. The characteristics of a regular decagon are:
- All sides are equal in length and all the angles are equal in the measure in a regular decagon.
- Each interior angle in regular decagon measures 144º, while each exterior angle measures 36º.
Irregular Decagon
An irregular decagon does not have equal sides and angles. At least two sides and angles are different in measurement. Look at the images given below showing irregular decagons.
Convex and Concave Decagons
Like any other polygon, decagons also can be convex and concave. A convex decagon bulges outward as all the interior angles are lesser than 180°. While concave decagons have indentations [a deep recess]. At least one of the interior angles is greater than 180° in concave decagons.
Simple and Complex Decagons
Simple decagons refer to decagons with no sides crossing themselves. They follow all of the above said regular decagon rules. While complex decagons refer to decagons that are self-intersecting and have additional interior spaces. They do not strictly follow any prescribed rules of decagons regarding their interior angles and their sums.
Properties of Decagon
Some of the important properties of decagons are listed here.
- The sum of the interior angles is 1440°.
- The sum of the measurements of the exterior angle is 360°.
- The central angle measures 36 degrees in the case of a regular decagon.
- There are 35 diagonals in a decagon.
- There are 8 triangles in a decagon.
Sum of the Interior Angles of Decagon
To find the sum of the interior angles of a decagon, first, divide it into triangles. There are eight triangles in a regular decagon. We know that the sum of the angles in each triangle is 180°. Thus, 180° × 8 = 1440°. Therefore, the sum of all the interior angles of a decagon is 1440°.
We know that the number of sides of a decagon is 10. Hence, we divide the total sum of the interior angles by 10
1440° ÷ 10 = 144°
Thus, one interior angle of a regular decagon shape is 144°. And, the sum of all the interior angles of a decagon is 1440°.
Measure of the Central Angles of a Regular Decagon
To find the measure of the central angle of a regular decagon, we need to draw a circle in the middle. A circle forms 360°. Divide this by ten, because a decagon has 10 sides. 360° ÷ 10 = 36°. Thus, the measure of the central angle of a regular decagon is 36°.
Decagon Diagonals
A diagonal is a line that can be drawn
from one vertex to another. The number of diagonals of a polygon is calculated by: n[n−3] ÷ 2. In decagon, n is the number of sides which is equal to 10, so n=10. Now we get,
n[n−3] ÷ 2 = 10[10−3] ÷ 2
Thus, the number of diagonals in a decagon is 35.
Decagon has 8 Triangles
By joining one vertex to the remaining vertices of the decagon, 8 triangles will be formed. By joining all the vertices independently to each other, then 80 triangles [8×10] will be formed. Look at the image given below, showing diagonals and triangles of a decagon.
Decagon Formula
Like other shapes, a regular decagon also has the formula to calculate perimeter and area. The formulas are mentioned below:
The formula to find the area of a decagon is \[\dfrac{5a^2}{2} \times \sqrt{5 + 2\sqrt{5}}\], where a is the measurement of the side-length of the decagon.
The formula to find the perimeter of a regular decagon is 10 times of a side or 10 × n, where n is the side-length of the decagon [as all sides are equal and total sides are 10]. In the case of an irregular decagon, we can simply add the side lengths to find the perimeter.
Important Notes
- A decagon has ten sides.
- The sum of the interior angles of a decagon is 1440°.
- The sum of the exterior angles of a decagon is 360°.
- A regular decagon has 35 diagonals.
- Decagons can be classified as regular, irregular, convex, concave, simple, and complex.
Related Articles
Check out these interesting articles to know more about decagon and its related topics.
- Area of decagon
- Dodecagon
- How many lines of symmetry does a regular decagon have?
FAQs on Decagon
What is a Decagon in Geometry?
A decagon can be recognized as a polygon with 10 sides, 10 interior angles, and 10 vertices. Decagon can be regular and irregular. Regular decagon has all equal sides. Irregular decagon has unequal sides.
How do you find the Area of a Decagon?
The formula to find the area of a regular decagon is 5a2/2 × \[\sqrt{5 + 2\sqrt{5}}\], where a is the measurement of the side of the decagon. If we know the measurement of the side length of a regular decagon, then we can use this formula to find the area. There is no formula to find the area of an irregular decagon, but we can divide it into triangles and try to find the area of all triangles and add them.
How Many Sides Does a Decagon have?
According to the properties of a decagon, the total number of sides is 10 made by joining 10 vertices. As there are only 10 sides and 10 vertices, therefore, the number of interiors and exteriors angles formed by all 10 sides are also 10.
How Many Vertices and Diagonals Do a Regular Decagon Have?
A regular decagon has 10 vertices and 35 diagonals. The diagonals of a decagon are found by using the formula n[n-3] ÷ 2, where n is a number of sides that is 10. So, 10[10-3] ÷ 2 = 35.
What are the Exterior Angles of a Regular Decagon?
The exterior angles of a decagon are the outside angles formed by two sides joined by one vertex of a decagon. Every exterior angle is adjacent to the corresponding interior angle and forms 180 degrees at the vertex of the decagon. In the regular decagon, all angles are equal. The sum of all exterior angles of the regular decagon is 360°. As the number of sides is 10 in a decagon. Therefore each exterior angle is equal to 36°[360° ÷ 10 =36°].
What is the Sum of the Interior Angles of a Decagon?
The sum of all the interior angles of a decagon is 1440°. There is a formula to find the sum of interior angles of the n-sided polygon [[n-2][180]] where n is equal to the number of sides. Thus, sum of interior angles of decagon = [10-2] × 180° = 1440°. Also, the measure of each interior angle in a regular decagon can be found by dividing the sum and the number of the total sides in the decagon ⇒1,440/10 = 144°.
What is a 10 Sided Shape Called?
A decagon is known as a ten-sided shape or polygon with ten vertices and ten angles. When any shape is formed by joining 10 sides, it is called a decagon.