The area of a regular polygon can be found using apothem and perimeter. Which formula computes the area given perimeter P and apothem a?

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Multiple Choice

The area of a regular polygon can be found using apothem and perimeter. Which formula computes the area given perimeter P and apothem a?

Dividing a regular polygon into congruent triangles using the apothem as the height shows how to combine both P and a. Each triangle has area (1/2) times its base (a side length s) times its height (the apothem a). With n such triangles, the total area is A = n*(1/2)*s*a = (1/2)*(n*s)*a. Since the perimeter P equals n*s, this becomes A = (1/2) P a. This is the correct relation because it sums the areas of all those triangles neatly. The other forms miss the factor of 1/2 or mix up how P and a relate to the area.

Dividing a regular polygon into congruent triangles using the apothem as the height shows how to combine both P and a. Each triangle has area (1/2) times its base (a side length s) times its height (the apothem a). With n such triangles, the total area is A = n*(1/2)sa = (1/2)(ns)a. Since the perimeter P equals ns, this becomes A = (1/2) P a. This is the correct relation because it sums the areas of all those triangles neatly. The other forms miss the factor of 1/2 or mix up how P and a relate to the area.

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