Question

formulas of area of regular shapes

Answer 1

A regular polygon is a 2-dimensional convex figure with congruent sides and angles equal in measure. Many polygons, such as quadrilaterals or triangles have simple formulas for finding their areas, but if you're working with a polygon that has more than four sides, then your best bet may be to use a formula that uses the shape's apothem and perimeter. With a little bit of effort, you can find the area of regular polygons in just a few minutes.

Part1Calculating the Area1Calculate the perimeter. The perimeter is the combined length of the outline of any two-dimensional figure. For a regular polygon, it can be calculated by multiplying the length of one side by the number of sides (n).[1]2Determine the apothem. The apothem of a regular polygon is the shortest distance from the center point to one of the sides, creating a right angle. This is a little trickier to calculate than the perimeter.The formula for calculating the length of the apothem is this: the length of the side (s) divided by 2 times the tangent (tan) of 180 degrees divided by the number of sides (n).3Know the correct formula. The area of any regular polygon is given by the formula: Area = (a x p)/2, where a is the length of the apothem and p is the perimeter of the polygon.4Plug the values of a and p in the formula and get the area. As an example, let's use a hexagon (6 sides) with a side (s) length of 10.The perimeter is 6 x 10 (n x s), equal to 60 (so p = 60).The apothem is calculated by its own formula, by plugging in 6 and 10 for n and s. The result of 2tan(180/6) is 1.1547, and then 10 divided by 1.1547 is equal to 8.66.The area of the polygon is Area = a x p / 2, or 8.66 multiplied by 60 divided by 2. The solution is an area of 259.8 units.Note as well, there are no parenthesis in the "Area" equation, so 8.66 divided by 2 multiplied by 60, will give you the same result, just as 60 divided by 2 multiplied by 8.66 will give you the same result.Part2Understanding the Concepts in a Different Way1Understand that a regular polygon can be thought of as a collection of triangles. Each side represents the base of a triangle, and there are as many triangles in the polygon as there are sides. Each of the triangles are equal in base length, height, and area.[2]2Remember the formula for the area of a triangle. The area of any triangle is 1/2 times the length of the base (which, in the polygon, is the length of a side) multiplied by the height (which is the same as the apothem in regular polygon).[3]3See the similarities. Again, the formula for a regular polygon is 1/2 times the apothem multiplied by the perimeter. The perimeter is just the length of one side multiplied the by the number of sides (n); for a regular polygon, n also represents the number of triangles that make up the figure. The formula, then, is nothing more than the area of a triangle multiplied by the number of triangles in the polygon.[4]