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How to find the area of a dodecagon?

Understand the Problem

The question is asking about the method to calculate the area of a dodecagon, which is a 12-sided polygon. To find this area, one can use the formula for the area of a regular dodecagon or design the approach based on the dodecagon's dimensions.

Answer

The area of a dodecagon with side length $s$ is $$ A = 3 \times (2 + \sqrt{3}) \times s^2 $$. For $s = 5$, the area is approximately 75.00 square units.
Answer for screen readers

The area of a regular dodecagon with side length $s$ is given by $$ A = 3 \times (2 + \sqrt{3}) \times s^2 $$

For $s = 5$, the area is approximately $A \approx 75.00$ square units.

Steps to Solve

  1. Identify the Area Formula for a Regular Dodecagon

A regular dodecagon (12-sided polygon) can have its area calculated using the formula: $$ A = 3 \times (2 + \sqrt{3}) \times s^2 $$ where $s$ is the length of one side of the dodecagon.

  1. Determine the Side Length

If not given, you will need the length of one side of the dodecagon ($s$). For example, if it is provided that the side length is 5 units, use that value in the formula.

  1. Substitute the Side Length into the Formula

Using the identified formula, substitute the side length into the equation. If $s = 5$, then substitute this value: $$ A = 3 \times (2 + \sqrt{3}) \times 5^2 $$

  1. Calculate the Square of the Side Length

Perform the calculation for $s^2$. For our example, $$ 5^2 = 25 $$

  1. Complete the Area Calculation

Now plug in the value of $s^2$ back into the area formula: $$ A = 3 \times (2 + \sqrt{3}) \times 25 $$

  1. Simplify the Equation

Calculate the numerical parts step by step:

  • First, calculate $2 + \sqrt{3}$.
  • Then multiply the result by 25.
  • Finally, multiply that product by 3.

The area of a regular dodecagon with side length $s$ is given by $$ A = 3 \times (2 + \sqrt{3}) \times s^2 $$

For $s = 5$, the area is approximately $A \approx 75.00$ square units.

More Information

The formula for the area of a regular dodecagon can also be derived using trigonometry by relating it to the sine function. Fun fact: the dodecagon is a shape that appears in various aspects of art and design, as well as in nature.

Tips

  • Forgetting to square the side length before multiplying.
  • Miscalculating the value of $2 + \sqrt{3}$.
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