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PolygonProperties”Circumcircle Incircle Radius and Apothem “Breaking Into TrianglesA Smaller TriangleMore Area FormulasA Table of ValuesGraphA polygon is a planeshape (twodimensional) with straight sides Examples include triangles quadrilaterals pentagons hexagons and so on So what can we know about regular polygons? First of all we can work out angles All the Exterior Angles of a polygon add up to 360° so Each exterior angle must be 360°/n (where nis the number of sides) Press play button to see Interior Angle = 180° − Exterior Angle We know theExterior angle = 360°/n so Interior Angle = 180° − 360°/n And now for some names Sounds quite musical if you repeat it a few times but they are just the names of the “outer” and “inner” circles (and each radius) that can be drawn on a polygon like this The “outside” circle is called a circumcircle and it connects all vertices (corner points) of the polygon The radius of the circumcircle is also the radiusof the polygon The “inside” circle is called an incircleand it just touches each side of the polygon at its midpoint The radius of the incircle is the apothemof the polygon (Not all polygons have those properties but triangles and regular polygons do) We can learn a lot about regular polygons by breaking them into triangles like this Notice that 1 the “base” of the triangle is one side of the polygon 2 the “height” of the triangle is the “Apothem” of the polygon Now the area of a triangleis half of the base times height so Area of one triangle = base × height / 2 = side × apothem / 2 To get the area of the whole polygon just add up the areas of all the little triangles (“n” of them) Area of Polygon = n× side × apothem / 2 And since the perimeter is all the sides = n × side we get Area of Polygon = perimeter × apothem / 2 By cutting the triangle in half we get this (Note The angles are in radians not degrees) The small triangle is rightangled and so we can use sine cosine and tangent to find how the side radius apothem and n(number of sides) are related There are a lot more relationships like those (most of them just “rearrangements”) but those will do for now We can use that to calculate the area when we only know the Apothem And there are 2 such triangles per side or 2n for the whole polygon Area of Polygon = n × Apothem2 × tan(π/n) When we don&#39t know the Apothem we can use the same formula but reworked for Radius or for Side Area of Polygon = ½ × n × Radius2 × sin(2 × π/n) Area of Polygon = ¼ × n × Side2 / tan(π/n) And here is a table of Side Apothem and Area compared to a Radius of “1” using the formulas we have worked out And here is a graph of the table above but with number of sides (“n”) from 3 to 30 Notice that as “n” gets bigger the Apothem is tending towards 1 (equal to the Radius) and that the Area is tending towards π= 314159 just like a circle What is the Side length tending towards?.