what is an equation for a regular pentagon?

Question

give an equation for a regular pentagon, whichever coordinate system u like:

Answer 1

There is no single equation. However, you could look at each of the five sides and construct a linear equation for each of them. Of course, the pentagon could have any center, angular orientation, and radius that you wanted. Assuming a pentagon with its center at the origin and a radius of 1 with one vertex at (0, 1) gives us someplace to start.

Look at the vertex in the first quadrant. Given that each pair of adjacent vertices forms an angle of 72 degrees with the origin (360 / 5 = 72), and one vertex is on the y-axis, the first quadrant vertex is at a polar coordinate of (r,theta) = (1, 18), which translates to (x,y) = (1*cos18, 1*sin18) = (cos18, sin18). With increments of 72 degrees, the other points are, in counterclockwise order, (cos90, sin90) = (1, 0), (cos162, sin162), (cos234, sin234), and (cos306, sin306). Not that the next point would be (cos378, sin378), which is equivalent to the (cos18, sin18) point we already identified.

Now, just calculate those values, and use the pairs of points to determine the equations of the lines. For example, (cos18, sin18) and (1, 0). (cos18, sin18) is (0.95, 0.31). The slope is (0 - 0.31) / (1 - 0.95) = -6.31, and we know that the y-intercept is 1, so the line is y = -6.31x + 1, for 0 < x < 0.95. For pairs that don't include a point on the y-axis, you might prefer to use the point-slope form, y - k = m(x - h) for slope m and a point (h, k).