EXTERIOR ANGLE OF A REGULAR OCTAGON

Example: What is the exterior edge of a continual octagon?

An octagon has 8 sides, so:

Interior Angles The internal Angle and also Exterior Angle room measured indigenous the exact same line, so they add up to 180°.

Interior edge = 180° − Exterior Angle

We recognize the Exterior angle = 360°/n, so:

Interior angle = 180° − 360°/n

Example: What is the interior angle of a consistent octagon?

A constant octagon has actually 8 sides, so:

Exterior edge = 360° / 8 = 45°

inner Angle = 180° − 45° = 135°

Interior Angle(of a continual octagon)

Or we can use:

Example: What room the interior and exterior angles of a consistent hexagon?

A regular hexagon has actually 6 sides, so:

Exterior angle = 360° / 6 = 60°

internal Angle = 180° − 60° = 120°

And now for some names:

"Circumcircle, Incircle, Radius and Apothem ..."

Sounds fairly musical if friend repeat the a few times, however they are simply the surname of the "outer" and also "inner" one (and each radius) that can be drawn on a polygon prefer this:

The "outside" one is called a circumcircle, and it connects every vertices (corner points) of the polygon.

The radius the the circumcircle is likewise the radius that the polygon.

The "inside" circle is dubbed an incircle and it simply touches each side the the polygon at its midpoint.

The radius the the incircle is the apothem the the polygon.

(Not every polygons have actually those properties, however triangles and regular polygon do).

Breaking into Triangles

We can learn a lot around regular polygons by break them right into triangles choose this:

Notice that:

Now, the area of a triangle is half of the basic times height, so:

Area that one triangle = base × elevation / 2 = side × apothem / 2

To get the area that the whole polygon, just add up the areas of every the tiny triangles ("n" the them):

Area of Polygon = n × side × apothem / 2

And because the perimeter is every the sides = n × side, we get:

Area the Polygon = perimeter × apothem / 2

A smaller sized Triangle

By cutting the triangle in fifty percent we get this:

The tiny triangle is right-angled and also so we deserve to use sine, cosine and tangent to find how the side, radius, apothem and n (number that sides) room related:

There space a lot more relationships choose those (most of them simply "re-arrangements"), however those will do for now.

More Area Formulas

We can use that to calculate the area once we only recognize the Apothem:

Triangle (or Trigon)	3		60°	1	1.732 (√3)	0.5	1.299 (¾√3)
Quadrilateral(or Tetragon)	4		90°	1	1.414 (√2)	0.707 (1/√2)	2
Pentagon	5		108°	1	1.176	0.809	2.378
Hexagon	6		120°	1	1	0.866 (½√3)	2.598 ((3/2)√3)
Heptagon (or Septagon)	7		128.571°	1	0.868	0.901	2.736
Octagon	8		135°	1	0.765	0.924	2.828 (2√2)
...
Pentacontagon	50		172.8°	1	0.126	0.998	3.133
(Note: worths correct to 3 decimal places only)

Graph

And here is a graph that the table above, yet with variety of sides ("n") native 3 come 30.

See more: Percentage Calculator: What Is 80 Percent Of 150, 80 Percent Of 150

Notice that together "n" it s okay bigger, the Apothem is tending towards 1 (equal come the Radius) and that the Area is tending towards π = 3.14159..., as with a circle.