It’s a good habit to get into to learn a new thing every day. And in keeping with that, here’s an interesting mathematical concept you might not be familiar with: “12-sided shape” or “Dodecagon.” What does that mean? We’re happy you asked. Here are its definition, types, and formulas, plus some geometry worksheets to help reinforce the concept.
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What is a 12-sided shape?
A dodecagon is a polygon with 12 sides. Essentially, a dodecagon is a polygon with 12 sides, 12 angles, and 12 vertices. The Greek words “dodeka” and “hedra,” which mean “twelve” and “seat/base,” are the origin of the 12-sided shape’s name.
Pentagons are 2-D objects having five straight edges that make five angles at the corners, as you will be aware if you have ever visited Washington, D.C. Dodecahedrons, which are three-dimensional entities with twelve flat sides that are all pentagonal in form, are pronounced “dow-deh-kuh-hee-druhns.”
Dodecagons can be either regular, in which case all of the internal angles and sides have the same measurement, or irregular, in which case the angles and sides have various dimensions.
Types of 12-sided shapes
Dodecagons come in a variety of shapes based on the size of their sides, angles, and other attributes. You might find these examples of 12-sided shapes interesting:
Regular Dodecagon
The regular dodecagon is the most typical kind of dodecagon. It features 12 equal-length sides, equal-angle corners, and vertices that are evenly spaced from the center. A 12-sided polygon with symmetry, the regular dodecagon.
Irregular Dodecagon
The irregular dodecagon has sides that are different forms and angles from the standard one. These dodecagons can come in a huge variety, but they always have twelve sides.
Concave Dodecagon
At least one line segment that can be drawn between the points on a concave dodecagon’s boundary but is outside of it defines the shape. The concave dodecagon also has at least one inner angle that is more than 180 degrees.
Convex Dodecagon
No line segment connecting any two locations on the boundary of a convex dodecagon’s 12-sided form polygon falls outside of it. Its internal angles are all less than 180 degrees.
Recommendation: To explore more shapes, check out our types of polygons guide.
Properties of a Dodecagon
These are Dodecagon’s properties:
- In a normal dodecagon, each inner angle is equal to 150°.
- In a regular dodecagon, each external angle is equal to 30°.
- In a dodecagon, there are 54 diagonals.
- The diagonals that are drawn from a dodecagon’s vertices can divide into a number of triangles. Ten triangles can be created inside of a dodecagon.
- A dodecagon’s inner angles add up to 1800°.
The following list of a dodecagon’s properties provides in-depth explanations of its angles, triangles, and diagonals.
Interior Angles of a 12-sided shape
A standard dodecagon has 150° for each inner angle. Using the following formula, this can be determined: 180n-360n
Where n is the polygon’s number of sides. n = 12 in a dodecagon. Put this value into the formula now. 180×(12)-360×12=150°
The formula (n – 2)×180° = (12 – 2)×180° = 1800° can be used to compute the total internal angle of a dodecagon.
Exterior Angles of a 12-sided shape
In a regular dodecagon, each external angle is equal to 30°. The external angle and inner angle combine to make a straight angle (180°) if we look at the illustration above. 180° – 150° = 30°, thus. Therefore, each external angle is 30° in length. A standard dodecagon’s external angles add up to 360°.
Diagonals of a 12-sided shape
The formula 1/2n(n-3), where n is the number of sides, can be used to determine how many separate diagonals can be formed from all of a dodecagon’s vertices. Here, n equals 12. In the formula, changing the values: 1/2 × n × (n-3) = 1/2 × 12 × (12-3) = 54
Hence, a dodecagon has 54 diagonals.
Triangles in a 12-sided shape
The diagonals that are drawn from a dodecagon’s vertices can divide into a number of triangles. The formula (n – 2), where n is the number of sides, may be used to determine how many triangles are produced by these diagonals. Here, n equals 12. So, 12 – 2 = 10. As a result, a dodecagon will be created from 10 triangles.
The key properties of a 12-sided shape that were previously mentioned are all recalled and listed in the table below.
| Properties | Values | |||
|---|---|---|---|---|
| Each Interior angle | 150° | |||
| Each Exterior angle | 30° | |||
| Number of diagonals | 54 | |||
| Number of triangles | 10 | |||
| Sum of the interior angles | 1800° |
The perimeter of a 12-side shape
A regular dodecagon’s perimeter can be calculated by adding up all of its sides, or by multiplying one side’s length by the entire number of sides. P = s x 12 can be used to indicate this, where s stands for the side length. Let’s say a standard dodecagon has sides that are 10 units wide. As a result, the circumference will be 10 x 12 or 120 units.
Area of a 12-side shape
The following equation may be used to calculate the area of a regular dodecagon:
A = 3 × ( 2 + √3 ) × s2
where A is the dodecagon’s area and s is the size of one of its sides.
For instance, if the side of a standard dodecagon is 10 units, its area will be:
A = 3(2+√3)102=1,119.615
Read more >> What is Rectangular Shape? Definition, Properties & Examples
What is the dodecagon’s practical application?
Although it might appear a little too difficult to recreate this beautiful 12-sided design, in reality, this is far from the case. Simply by paying more attention, we may see the surrounding dodecagons on anything from different currencies to historic structures all across the world.
The letters E, H, and X, for example, all exhibit dodecagonal shapes when written in block capitals. Even though it’s not a standard dodecagon, a cross is still a dodecagon. There are several significant buildings that also have the standard dodecagon.
A dodecagon is, for instance, the Vera Cruz chapel in Segovia, Spain, which dates to the early thirteenth century. The Porta di Venere (Venus’ Gate), in Spello, Italy, has two towers that are arranged in the shape of a dodecagon.
Numerous nations across the world have coins with a 12-sided design. For instance, the 2017-introduced British One Pound Coin is a dodecagon. This form can also be found on Australian 50-cent coins and Croatian 25-kuna coins.
Recommendation: Explore more real-life geometry examples like coins, buildings, and lettering here.
Key takeaway
When handling issues involving a dodecagon, the following points should be kept in mind.
- A dodecagon is a polygon with 12 sides, 12 angles, and 12 vertices.
- The interior angles of a dodecagon add up to 1800°.
- The perimeter of a 12-side shape: P = s × 12.
- Area of a 12-side shape: A = 3(2+√3)s2
FAQs
1. Are dodecagons considered regular polygons?
If a dodecagon has 12 equal-length sides and 12 equal-sized interior angles, it can be defined as a regular polygon.
2. How many diagonals are there in a dodecagon?
One dodecagon has 54 diagonals.
3. How many triangles are there in a dodecagon?
By using the diagonals that are traced from its vertices, a dodecagon may be divided into a number of triangles. A dodecagon can be divided into 10 different triangular shapes.
4. What are the dodecagon’s total exterior angles?
No matter how many sides a polygon has, the total of its exterior angles is always equal to 360°. As a result, the total of the outer angles is 360° even in a dodecagon.
5. What are the dodecagon’s total interior angles?
A dodecagon’s internal angles add up to 1800 °.
6. What is one exterior angle of a dodecagon?
A dodecagon’s outside angles are each equal to 30°.
7. What properties do regular dodecagons have?
Each angle inside a standard decagon is equal to 150°, whereas each angle outside is equal to 30°. Its inside angles total 1800°, whilst its external angles total 360°.
If you are planning to teach your kids about this essential topic, you can make your own collections of dodecagon worksheets using our Worksheet Maker.
