the area of a regular polygon varies jointly as its apothem and its perimeter. express the area in terms pf apothem a and the perimeter p

A = k a p

side s = p/n
base of triangle of one section = s = p/n
1/2 base = p/2n
if n sides
then area = n (a) p/2n) = a p/2

To express the area of a regular polygon in terms of its apothem (a) and perimeter (p), we need to find the relationship between these variables.

In a regular polygon, the apothem represents the distance from the center of the polygon to any side, and the perimeter represents the total length of all sides.

The formula for the area of a regular polygon is given by:

Area = 1/2 * apothem * perimeter

Since the area varies jointly with the apothem and perimeter, we can rewrite the formula by introducing a constant of variation (k):

Area = k * apothem * perimeter

Now, to express the area in terms of the apothem (a) and perimeter (p), we need to determine the value of the constant of variation (k). This can be done by using the known values of the apothem and perimeter for a specific regular polygon.

Let's assume we have a regular polygon with apothem 'a' and perimeter 'p'. We can substitute these values into the formula:

Area = k * a * p

To find the constant of variation (k), we can rearrange the formula:

k = Area / (a * p)

Now, we can express the area of a regular polygon in terms of the apothem (a) and perimeter (p) by substituting the value of the constant of variation (k):

Area = (Area / (a * p)) * a * p

Simplifying the expression:

Area = Area

Therefore, the area of a regular polygon in terms of its apothem (a) and perimeter (p) is simply the given area itself.