Question

Find the area of a dodecagon (or a 12-gon) that has 4 inch
sides.

Answer 1

The area of a dodecagon (12-gon) with 4-inch sides can be found by dividing it into triangles and using trigonometric formulas.

Step-by-step explanation:

The area of a regular dodecagon can be found by dividing it into triangles and using trigonometric formulas. Each triangle can be divided into two right triangles by drawing a height from one of the vertices to the center of the dodecagon. The length of the side of the dodecagon is the hypotenuse of each right triangle. The height of each right triangle can be found using the formula h = a * sin(15°), where a is the side length and h is the height. The area of each right triangle is then A = 0.5 * a * h. Since there are 12 triangles in a dodecagon, you can multiply the area of one triangle by 12 to find the total area of the dodecagon.

Let's calculate the area step by step:

1. Find the height of one triangle: h = 4 * sin(15°) inches.
2. Calculate the area of one triangle: A = 0.5 * 4 * 4 * sin(15°) square inches.
3. Multiply the area of one triangle by 12: Total area = 12 * A square inches.

Thus, the area of the dodecagon is 12 * A square inches.