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Apothemof a Regular Polygon
Definition: A line segment from the center of a regular polygon to the midpoint of a side.
Try this
Adjust the polygon below by dragging any orange dot, or alter the number of sides. Note the behavior of the apothem line shown in blue.

The apothem is also the radius of the
incircle of the polygon .
For a polygon of n sides, there are n possible apothems,
all the same length of course.
The word apothem can refer to the line itself, or the length of that line.
So you can correctly say ‘draw the apothem’ and ‘the apothem is 4cm’.

Each formula below shows how to find the length of the apothem of a regular polygon.
Use the formula that uses the facts you are given to start.

Apothem given the length of a side.

By definition, all sides of a regular polygon are equal in length. If you know the length of one of the sides, the apothem length is given by the formula:

where:
s  is the length of any side
n  is the number of sides
tan  is the tangent function calculated in degrees (see Trigonometry Overview )

Apothem given the radius (circumradius)

If you know the radius (distance from the center to a vertex ):

where
r  is the radius (circumradius) of the polygon
n  is the number of sides
cos  is the cosine function calculated in degrees (see Trigonometry Overview )

Irregular Polygons

Since irregular polygons have no center, they have no apothem. In the figure above, uncheck the “regular” checkbox and note how there can be no center or apothem.

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Other polygon topics

General

  • Polygon general definition
  • Quadrilateral
  • Regular polygon
  • Irregular polygon
  • Convex polygons
  • Concave polygons
  • Polygon diagonals
  • Polygon triangles
  • Apothem of a regular polygon
  • Polygon center
  • Radius of a regular polygon
  • Incircle of a regular polygon
  • Incenter of a regular polygon
  • Circumcircle of a polygon
  • Parallelogram inscribed in a quadrilateral

Types of polygon

  • Square
  • Diagonals of a square
  • Rectangle
  • Diagonals of a rectangle
  • Golden rectangle
  • Parallelogram
  • Rhombus
  • Trapezoid
  • Trapezoid median
  • Trapezium
  • Kite
  • Inscribed (cyclic) quadrilateral
    • Inscribed quadrilateral interior angles
    • Inscribed quadrilateral area
    • Inscribed quadrilateral diagonals

Area of various polygon types

  • Regular polygon area
  • Irregular polygon area
  • Rhombus area
  • Kite area
  • Rectangle area
  • Area of a square
  • Trapezoid area
  • Parallelogram area

Perimeter of various polygon types

  • Perimeter of a polygon (regular and irregular)
  • Perimeter of a triangle
  • Perimeter of a rectangle
  • Perimeter of a square
  • Perimeter of a parallelogram
  • Perimeter of a rhombus
  • Perimeter of a trapezoid
  • Perimeter of a kite

Angles associated with polygons

  • Exterior angles of a polygon
  • Interior angles of a polygon
  • Relationship of interior/exterior angles
  • Polygon central angle

Named polygons

  • Tetragon, 4 sides
  • Pentagon, 5 sides
  • Hexagon, 6 sides
  • Heptagon, 7 sides
  • Octagon, 8 sides
  • Nonagon Enneagon, 9 sides
  • Decagon, 10 sides
  • Undecagon, 11 sides
  • Dodecagon, 12 sides
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RELATED ARTICLES

  • How to Calculate the Area of a Regular Polygon

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  • Interior and Exterior Angles of a Polygon

  • How to Determine the Length of an Arc

  • Properties of Rhombuses, Rectangles, and Squares

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  4. How to Calculate the Area of a Regular Polygon

How to Calculate the Area of a Regular Polygon

Related Book

Geometry For Dummies, 2nd Edition

By Mark Ryan

A regular polygon is equilateral (it has equal sides) and equiangular (it has equal angles). To find the area of a regular polygon, you use an apothem — a segment that joins the polygon’s center to the midpoint of any side and that is perpendicular to that side (segment HM in the following figure is an apothem).

image0.jpg

You use the following formula to find the area of a regular polygon:

image1.png

So what’s the area of the hexagon shown above?

You need the perimeter, and to get that you need to use the fact that triangle OMH is a triangle (you deduce that by noticing that angle OHG makes up a sixth of the way around point H and is thus a sixth of 360 degrees, or 60 degrees; and then that angle OHM is half of that, or 30 degrees).

In a triangle, the long leg is times as long as the short leg, so that gives a length of 10. is twice that, or 20, and thus the perimeter is six times that or 120. Now just plug everything into the area formula:

You’re done.

You could use this regular polygon formula to figure the area of an equilateral triangle (which is the regular polygon with the fewest possible number of sides), but there are two other ways that are much easier. First, you can get the area of an equilateral triangle by just noting that it’s made up of two triangles. You can see how this works with triangle OHG in the figure above. Second, the equilateral triangle has its own area formula so that’s a really easy way to go assuming you’ve got some available space on your gray matter hard drive:

Area of an equilateral triangle: Here’s the formula.

image2.png