Summarize this article:
Last updated on September 9, 2025
The area is the space inside the boundaries of a two-dimensional shape or surface. There are different formulas for finding the area of various shapes/figures. These are widely used in architecture and design. In this section, we will find the area of a dodecagon.
A dodecagon is a twelve-sided polygon with twelve equal angles. It can be regular or irregular, but here we focus on the regular dodecagon, where all sides and angles are equal. The area of a dodecagon is the total space it encloses.
To find the area of a regular dodecagon, we can use the formula: ( text{Area} = 3 × (2 + sqrt{3}) × s2 ), where ( s ) is the length of a side. This formula is derived from dividing the dodecagon into 12 isosceles triangles and calculating the area of each triangle.
Derivation of the formula: A regular dodecagon can be divided into 12 equal isosceles triangles. Each triangle has a base of length ( s ) and a vertex angle of 30 degrees. The area of one triangle is ( frac{1}{2} × s × s ×sin(30circ) ).
Since (sin(30circ) = frac{1}{2}), the area of one triangle becomes ( frac{s2}{4} ). Multiplying by 12, the area of the dodecagon is ( 12 × frac{s2}{4} ×(2 + sqrt{3}) ).
Therefore, the area of the dodecagon = ( 3 × (2 + sqrt{3}) ×s2 ).
To find the area of a dodecagon, you can use the formula derived above. This involves knowing the length of one side of the dodecagon.
For example, if the side length is 10 cm, the area of the dodecagon is calculated as follows: ( text{Area} = 3 × (2 + sqrt{3}) × 102 ) ( = 3 × (2 +sqrt{3}) × 100 ) ( approx 936.36 text{ cm}2 ).
We measure the area of a dodecagon in square units.
The measurement depends on the system used: In the metric system, the area is measured in square meters (m²), square centimeters (cm²), and square millimeters (mm²).
In the imperial system, the area is measured in square inches (in²), square feet (ft²), and square yards (yd²).
A dodecagon is a regular polygon with 12 sides, and its area can be calculated using the specific formula for regular polygons. The key is knowing the side length.
Here are some considerations: - Use the formula when the side length is known: ( text{Area} = 3 ×(2 + sqrt{3}) × s2 ).
For irregular dodecagons, different methods like decomposition into triangles or other polygons may be needed.
To ensure accurate calculations for the area of a dodecagon, consider the following tips and tricks:
It's common to make mistakes when calculating the area of a dodecagon. Let's review some frequent errors and how to avoid them.
A regular dodecagon-shaped garden has a side length of 8 m. What will be the area?
We will find the area as approximately 618.18 m².
Here, the side length ( s ) is 8 m.
The area of the dodecagon is calculated as: ( text{Area} = 3 × (2 + sqrt{3}) × 82 ) ( = 3 × (2 + sqrt{3}) × 64 ) ( approx 618.18 text{ m}2 ).
What will be the area of a dodecagon if the side length is 12 cm?
We will find the area as approximately 1296.72 cm².
The side length ( s ) is 12 cm.
Using the formula: ( text{Area} = 3 × (2 + sqrt{3}) × 122 ) ( = 3 × (2 + sqrt{3}) × 144 ) ( approx 1296.72 text{ cm}2 ).
The area of a dodecagon is approximately 2187.44 m², and one side is 15 m. Verify the calculation.
We verify the area as approximately 2187.44 m².
Given side length ( s = 15 ) m: ( text{Area} = 3 × (2 + sqrt{3}) × 152 ) ( = 3 × (2 + sqrt{3}) × 225 ) ( approx 2187.44 text{ m}2 ).
Find the area of the dodecagon if its side length is 6 cm.
We will find the area as approximately 277.12 cm².
The side length ( s ) is 6 cm.
Using the formula: ( text{Area} = 3 ×(2 + sqrt{3}) × 62 ) ( = 3 × (2 + sqrt{3}) × 36 ) ( approx 277.12 text{ cm}2 ).
Help Sarah find the area of a dodecagon if the side is 20 m.
We will find the area as approximately 3745.44 m².
The side length ( s ) is 20 m.
Calculate the area: ( text{Area} = 3 ×(2 +sqrt{3}) × 202 ) ( = 3 × (2 + sqrt{3}) ×400 ) ( approx 3745.44 text{ m}2 ).
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
: She has songs for each table which helps her to remember the tables