Geometry - Decagon
- What is a Decagon?
- Definition of decagon
- What is a regular decagon?
- Properties of a regular decagon
- Other types of decagons
- Solved examples
- Practice problems
- Frequently Asked Questions
What is a Decagon?
To understand a decagon, first, we should understand what a polygon is. A polygon is a two-dimensional closed figure that has several straight sides and angles. A polygon with four sides is called a quadrilateral, whereas a polygon with five sides is called a pentagon.
Similarly, a polygon with `10` sides is called a decagon. The word “Decagon” is a Greek word formed from two words, “Deca” and “gon”. “Deca” means ten and “gon” means angles or corners. So, we can define the decagon as a polygon having `10` sides and consequently `10` angles.
Can you identify the number of sides in the above polygons and name the polygons?
Definition of decagon
A decagon is defined as a polygon having `10` sides with `10` corners or angles. It is a geometrical shape having `10` sides and consequently `10` interior angles.
Let us visualize a decagon with the image below.
In the above image, you can see that the number of sides is `10` which are `AB, ``BC,`` CD,`` DE,`` EF,`` FG,`` GH,`` HI,`` IJ,` and `JA`. And there are `10` corners or angles which can be identified as `∠ABC,`` ∠BCD,`` ∠CDE,`` ∠DEF,`` ∠EFG,`` ∠FGH,`` ∠GHI,`` ∠HIJ,`` ∠IJA,` and `∠JAB`.
Besides `10` sides you can also notice the `10` vertices which are `A,`` B,`` C,`` D,`` E,`` E,`` F,`` G,`` H,`` I,` and `J`.
What is a regular Decagon?
A decagon is called a regular decagon if all its sides and interior angles are equal. Regular decagons hold the property of symmetry as well. Regular decagons can be found in applications such as geometrical structures, architectural elements, decorative ornaments, etc.
Properties of a regular decagon
Let us learn the properties of a regular decagon one by one.
- Number of sides: A decagon has ten sides.
- Number of angles: A decagon has `10` interior and `10` exterior angles.
- Interior angles: As shown in the image below the interior angles are the inside angles of a decagon. In a regular decagon, each interior angle measures `144°`.
- Sum of all interior angles: The sum of all the interior angles in a decagon is `1440°`.
Below is the formula to calculate the sum of all the interior angles of any polygon.
The sum of interior angles of any polygon`=(n-2)\times 180°`
where `n` is the number of sides in a polygon.
If we put `n=10` for a decagon, then
Sum of interior angles of a decagon`=(10-2)\times 180°=1440°`
Therefore, each interior angle of a regular decagon `=(1440)/(10)=144°`
- Exterior angles: The exterior angles are the angles outside a decagon as shown in the image above. Each exterior angle measures `36°` in a regular decagon and the sum of all the exterior angles in a decagon is `360°`.
- Side length measure: In a regular decagon, all the sides are of equal lengths.
- Symmetry: A regular decagon depicts a symmetry of order `10`, which signifies that if we rotate a regular decagon by one-tenth of `360°`, it will appear the same no matter how many times you rotate.
- Diagonal: A regular decagon has a total of `35` diagonals.
A diagonal is a line that is formed by joining any two non-adjacent vertices in a polygon. For a polygon number of diagonals can be calculated by the formula
Number of diagonals in a polygon`=\frac{n(n-3)}{2}`
where n is the number of sides of the polygon.
For a decagon `n=10`
Number of diagonals in a decagon`=(10(10-3))/2=(10\times 7)/2=35`
- Central angle: A central angle in a decagon is an angle that is formed by the intersection of diagonals at the center of the decagon. Each central angle of a regular decagon has the same measurement as all the sides of a regular decagon are of the same length.
The central angle of a polygon `= \frac{360}{\text{No. of sides }}`
The central angle of a decagon `=360/10 =36°`
- Area of a regular decagon: The area for a regular decagon can be calculated by the formula.
Area of a regular decagon `= 5/2 s^2\sqrt{5+2\sqrt{5}}`
Where `s` is the side length of the regular decagon.
- Perimeter: The perimeter is the sum of all the sides of a decagon.
For a regular decagon, it can be calculated by the formula,
The perimeter of a regular decagon`= 10s` where `s` is the side length of the regular decagon.
Other Types of Decagons
- Irregular Decagon: An irregular decagon is a decagon that may have different measures of its sides and angles.
- Convex Decagon: A decagon is called a convex decagon when all its interior angles have a measure of less than `180°` and all its diagonals lie inside the decagon.
- Concave Decagon: If in a decagon one or more of its interior angles are greater than `180°` and one or more of its diagonals lie outside, then it will be called a concave decagon.
Note: An irregular decagon can be either convex or concave.
Solved Examples
Q`1`. Find out the perimeter of a regular decagon with a side length of `7` cm.
Solution:
The perimeter of a decagon is given by `10s`, `s` being the side length.
Here `s =7` cm, then the perimeter of the decagon `= 10 \times 7 = 70` cm.
Q`2`. Calculate the length of the side of a regular decagon, if the perimeter of the decagon is `70` cm.
Solution:
Here, the perimeter of the decagon `= 70` cm
Also, perimeter of the decagon `=` Side length `\times 10`
Therefore,
Side length `= \frac{\text{Perimeter of the decagon}}{10}=(70)/(10)=7` cm
Q`3`. What is the measure of each interior angle of a regular decagon?
Solution:
Sum of all the interior angles of a regular decagon `= 1440°`.
Number of sides `= 10`
Measure of each interior angle `=\frac{\text{Sum of all interior angles} }{\text{Number of sides}}= (1440°)/(10) = 144°`
Practice Problems
Q`1`. How many diagonals does a decagon have?
- `25`
- `35`
- `45`
- `55`
Answer: b
Q`2`. Can a regular decagon be a convex decagon?
- Yes
- No
Answer: a
Q`3`. We know that by joining the lines from the center of a regular decagon to the vertices, triangles are formed. What type of triangles are these?
- Isosceles triangles
- Scalene triangles
- Equilateral triangles
- Right angle triangle
Answer: a
Q`4`. The angle of rotational symmetry in a regular decagon is _____.
- `72°`
- `36°`
- `144°`
- `18°`
Answer: b
Frequently Asked Questions
Q`1`. Do we have names for polygons with `5, 6, 7, 8, 9` etc. sides?
Answer: Yes, we do have names for all these types of polygons. Here is the list of some of the polygons.
Q`2`. Is there any history associated with decagons? Does it have any cultural significance as well?
Answer: Yes, in some cultures decagon is a symbol of completeness, oneness representing a tenfold symmetry.
Q`3`. Are there any practical cases of a regular decagon?
Answer: Yes, coasters, coins, drums, and watches are some of the common examples where we use the shape of a regular decagon.
Q`4`. Are decagons more significant than other shapes?
Answer: Although decagons are very important shapes in geometry and are very much used in architectural and engineering applications, their usage is superseded by more commonly used polygons like triangles, squares, rectangles, etc.