Like understanding any mathematical principle, while getting understanding geometry related to apothem, how the mathematical concept was derived, attributes of apothem that are used in formula to calculate it.
Apothem is the important mathematical concept that is used to know the perpendicular distance of any side of regular polygon from its center point. Regular polygon is two-dimensional plane geometric shape with all straight sides congruent to each other and all angles congruent to each other. In other words, regular polygon is both equiangular and equilateral. The examples of regular polygon are squares, quadrilaterals, equilateral triangles, hexagons, pentagon etc. Hence, the primary two conditions to apply apothem are:
- All sides should be equal (congruent)
- All angles should be equal (congruent)
AP calculus students often face difficult to understand this basic rule.
In mathematical term, apothem is a line segment from the center of the regular polygon drawn to the midpoint of one of its sides. In other words, it is line segment drawn from the center of the polygon perpendicular to one of its sides. The apothem length remains same for regular polygon as the length from the center of the polygon to any of its sides are congruent.
Apothem formula and its attributes
Apothem a = s / 2 tan (180 / n)
Or
a = R cos (180 / n)
s = sides of polygon
n = number of sides
tan = tangent calculated in degrees
An Apothem of regular polygon is the radius of inscribed circle and is a small distance from any sides of the polygon to its center.
Apothem is used to calculate the area of polygon of the side length ‘s’ which means the area will be apothem multiplied by half of the perimeter because ns = p.
A = nsa / 2
= pa / 2
The formula to calculate area A of any n-sided polygon is derived by dividing n-sided polygon into n isosceles congruent triangles and height of each triangle will be apothem and area of triangle will be base times to the height.
Irregular polygons do not have any center point, its sides are unequal and angles are also unequal which means they do not have apothem