Human: How can I find the area of a equilateral triangle (3 sides) given only one of it's side lengths.

Assistant: The area of a triangle *A* = *hb* / 2. *b* is the base, or the length of one of its sides. *h* is the height, or the distance between the vertex opposite to the chosen side and that side. For an equilateral triangle with a side length of *a*, the base is always *a*. As all of its interior angles are 60°, the height is *a* sin60° = √3 *a* / 2.
Therefore, the area *A* = *hb* / 2 = √3 *a* ^2 / 4.

Human: That is correct. What about the area for any regular convex polygon, given any one of it's sides. Is it reasonable to find this?

Assistant:  Human: How can I find the area of a equilateral triangle (3 sides) given only one of it's side lengths.

Assistant: The area of a triangle *A* = *hb* / 2. *b* is the base, or the length of one of its sides. *h* is the height, or the distance between the vertex opposite to the chosen side and that side. For an equilateral triangle with a side length of *a*, the base is always *a*. As all of its interior angles are 60°, the height is *a* sin60° = √3 *a* / 2.
Therefore, the area *A* = *hb* / 2 = √3 *a* ^2 / 4.

Human: That is correct. What about the area for any regular convex polygon, given any one of it's sides. Is it reasonable to find this?

Assistant:  Human: How can I find the area of a equilateral triangle (3 sides) given only one of it's side lengths.

Assistant: The area of a triangle *A* = *hb* / 2. *b* is the base, or the length of one of its sides. *h* is the height, or the distance between the vertex opposite to the chosen side and that side. For an equilateral triangle with a side length of *a*, the base is always *a*. As all of its interior angles are 60°, the height is *a* sin60° = √3 *a* / 2.
Therefore, the area *A* = *hb* / 2 = √3 *a* ^2 / 4.

Human: That is correct. What about the area for any regular convex polygon, given any one of it's sides. Is it reasonable to find this?

Assistant:  Human: How can I find the area of a equilateral triangle (3 sides) given only one of it's side lengths.

Assistant: The area of a triangle *A* = *hb* / 2. *b* is the base, or the length of one of its sides. *h* is the height, or the distance between the vertex opposite to the chosen side and that side. For an equilateral triangle with a side length of *a*, the base is always *a*. As all of its interior angles are 60°, the height is *a* sin60° = √3 *a* / 2.
Therefore, the area *A* = *hb* / 2 = √3 *a* ^2 / 4.

Human: That is correct. What about the area for any regular convex polygon, given any one of it's sides. Is it reasonable to find this?

Assistant:  Human: How can I find the area of a equilateral triangle (3 sides) given only one of it's side lengths.

Assistant: The area of a triangle *A* = *hb* / 2. *b* is the base, or the length of one of its sides. *h* is the height, or the distance between the vertex opposite to the chosen side and that side. For an equilateral triangle with a side length of *a*, the base is always *a*. As all of its interior angles are 60°, the height is *a* sin60° = √3 *a* / 2.
Therefore, the area *A* = *hb* / 2 = √3 *a* ^2 / 4.

Human: That is correct. What about the area for any regular convex polygon, given any one of it's sides. Is it reasonable to find this?

Assistant:  Human: How can I find the area of a equilateral triangle (3 sides) given only one of it's side lengths.

Assistant: The area of a triangle *A* = *hb* / 2. *b* is the base, or the length of one of its sides. *h* is the height, or the distance between the vertex opposite to the chosen side and that side. For an equilateral triangle with a side length of *a*, the base is always *a*. As all of its interior angles are 60°, the height is *a* sin60° = √3 *a* / 2.
Therefore, the area *A* = *hb* / 2 = √3 *a* ^2 / 4.

Human: That is correct. What about the area for any regular convex polygon, given any one of it's sides. Is it reasonable to find this?

Assistant:  Human: How can I find the area of a equilateral triangle (3 sides) given only one of it's side lengths.

Assistant: The area of a triangle *A* = *hb* / 2. *b* is the base, or the length of one of its sides. *h* is the height, or the distance between the vertex opposite to the chosen side and that side. For an equilateral triangle with a side length of *a*, the base is always *a*. As all of its interior angles are 60°, the height is *a* sin60° = √3 *a* / 2.
Therefore, the area *A* = *hb* / 2 = √3 *a* ^2 / 4.

Human: That is correct. What about the area for any regular convex polygon, given any one of it's sides. Is it reasonable to find this?

Assistant:  Human: How can I find the area of a equilateral triangle (3 sides) given only one of it's side lengths.

Assistant: The area of a triangle *A* = *hb* / 2. *b* is the base, or the length of one of its sides. *h* is the height, or the distance between the vertex opposite to the chosen side and that side. For an equilateral triangle with a side length of *a*, the base is always *a*. As all of its interior angles are 60°, the height is *a* sin60° = √3 *a* / 2.
Therefore, the area *A* = *hb* / 2 = √3 *a* ^2 / 4.

Human: That is correct. What about the area for any regular convex polygon, given any one of it's sides. Is it reasonable to find this?

Assistant:  Human: How can I find the area of a equilateral triangle (3 sides) given only one of it's side lengths.

Assistant: The area of a triangle *A* = *hb* / 2. *b* is the base, or the length of one of its sides. *h* is the height, or the distance between the vertex opposite to the chosen side and that side. For an equilateral triangle with a side length of *a*, the base is always *a*. As all of its interior angles are 60°, the height is *a* sin60° = √3 *a* / 2.
Therefore, the area *A* = *hb* / 2 = √3 *a* ^2 / 4.

Human: That is correct. What about the area for any regular convex polygon, given any one of it's sides. Is it reasonable to find this?

Assistant:  Human: How can I find the area of a equilateral triangle (3 sides) given only one of it's side lengths.

Assistant: The area of a triangle *A* = *hb* / 2. *b* is the base, or the length of one of its sides. *h* is the height, or the distance between the vertex opposite to the chosen side and that side. For an equilateral triangle with a side length of *a*, the base is always *a*. As all of its interior angles are 60°, the height is *a* sin60° = √3 *a* / 2.
Therefore, the area *A* = *hb* / 2 = √3 *a* ^2 / 4.

Human: That is correct. What about the area for any regular convex polygon, given any one of it's sides. Is it reasonable to find this?

Assistant:  Human: How can I find the area of a equilateral triangle (3 sides) given only one of it's side lengths.

Assistant: The area of a triangle *A* = *hb* / 2. *b* is the base, or the length of one of its sides. *h* is the height, or the distance between the vertex opposite to the chosen side and that side. For an equilateral triangle with a side length of *a*, the base is always *a*. As all of its interior angles are 60°, the height is *a* sin60° = √3 *a* / 2.
Therefore, the area *A* = *hb* / 2 = √3 *a* ^2 / 4.

Human: That is correct. What about the area for any regular convex polygon, given any one of it's sides. Is it reasonable to find this?

Assistant:  Human: How can I find the area of a equilateral triangle (3 sides) given only one of it's side lengths.

Assistant: The area of a triangle *A* = *hb* / 2. *b* is the base, or the length of one of its sides. *h* is the height, or the distance between the vertex opposite to the chosen side and that side. For an equilateral triangle with a side length of *a*, the base is always *a*. As all of its interior angles are 60°, the height is *a* sin60° = √3 *a* / 2.
Therefore, the area *A* = *hb* / 2 = √3 *a* ^2 / 4.

Human: That is correct. What about the area for any regular convex polygon, given any one of it's sides. Is it reasonable to find this?

Assistant:  Human: How can I find the area of a equilateral triangle (3 sides) given only one of it's side lengths.

Assistant: The area of a triangle *A* = *hb* / 2. *b* is the base, or the length of one of its sides. *h* is the height, or the distance between the vertex opposite to the chosen side and that side. For an equilateral triangle with a side length of *a*, the base is always *a*. As all of its interior angles are 60°, the height is *a* sin60° = √3 *a* / 2.
Therefore, the area *A* = *hb* / 2 = √3 *a* ^2 / 4.

Human: That is correct. What about the area for any regular convex polygon, given any one of it's sides. Is it reasonable to find this?

Assistant:  Human: How can I find the area of a equilateral triangle (3 sides) given only one of it's side lengths.

Assistant: The area of a triangle *A* = *hb* / 2. *b* is the base, or the length of one of its sides. *h* is the height, or the distance between the vertex opposite to the chosen side and that side. For an equilateral triangle with a side length of *a*, the base is always *a*. As all of its interior angles are 60°, the height is *a* sin60° = √3 *a* / 2.
Therefore, the area *A* = *hb* / 2 = √3 *a* ^2 / 4.

Human: That is correct. What about the area for any regular convex polygon, given any one of it's sides. Is it reasonable to find this?

Assistant:  Human: How can I find the area of a equilateral triangle (3 sides) given only one of it's side lengths.

Assistant: The area of a triangle *A* = *hb* / 2. *b* is the base, or the length of one of its sides. *h* is the height, or the distance between the vertex opposite to the chosen side and that side. For an equilateral triangle with a side length of *a*, the base is always *a*. As all of its interior angles are 60°, the height is *a* sin60° = √3 *a* / 2.
Therefore, the area *A* = *hb* / 2 = √3 *a* ^2 / 4.

Human: That is correct. What about the area for any regular convex polygon, given any one of it's sides. Is it reasonable to find this?

Assistant:  Human: How can I find the area of a equilateral triangle (3 sides) given only one of it's side lengths.

Assistant: The area of a triangle *A* = *hb* / 2. *b* is the base, or the length of one of its sides. *h* is the height, or the distance between the vertex opposite to the chosen side and that side. For an equilateral triangle with a side length of *a*, the base is always *a*. As all of its interior angles are 60°, the height is *a* sin60° = √3 *a* / 2.
Therefore, the area *A* = *hb* / 2 = √3 *a* ^2 / 4.

Human: That is correct. What about the area for any regular convex polygon, given any one of it's sides. Is it reasonable to find this?

Assistant: 
It is reasonable, and can be done with a simple formula A = N * s * a / 2 for a regular convex polygon with N sides of length s. a is the shortest distance from the center point to one of the sides, known as the apothem. This formula can be viewed as first dividing the regular convex polygon into N identical triangles by connecting its center to its vertices. The area of each triangle is s * a / 2, where s is the base and a is the height. The total area of the triangles is N * s * a / 2.
Within each triangle, If we draw a perpendicular line from the center of the polygon to the base, we get two identical right triangles. The angle with the polygon's center as the vertex is 180°/N, and the length of the side opposite to it is s / 2. Therefore, the side adjacent to it, or the apothem, is s * cot(180°/N) / 2, and the area of the regular convex polygon A = N * s^2 * cot(180°/N) / 4. The interior angles of a convex polygon are less than 180°.
Regularly, a polygon is firmly convex, if each line segment with two nonadjacent vertices of the polygon is strictly internal to the polygon but on its endpoints.
A regular convex polygon is a polygon where each side is of the same length, and all the interior angles are equal and less than 180 degrees.
The vertices and sides are evenly spread around a central point.

If the convex polygon has vertices, then the formula for the area of the convex polygon is: (number of sides \* length of one side \* apothem)/2

the length of apothem is given as the 1/(2 \* tan(180 / number of sides))