Polygon

A polygon is a aircraft shape (two-dimensional) with right sides. Examples incorporate triangles, quadrilaterals, pentagons, hexagons and also so on.

You are watching: Exterior angle of a regular octagon

Regular

A "Regular Polygon" has:


Otherwise it is irregular.

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Regular PentagonIrregular Pentagon

Here us look at Regular Polygons only.

Properties

So what can we know around regular polygons? very first of all, we deserve to work out angles.

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Exterior Angle

The Exterior angle is the edge between any type of side of a shape, and also a line prolonged from the next side.


AlltheExterior angles of a polygon include up to 360°, so:

Each exterior angle should be 360°/n

(where n is the number of sides)

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Exterior Angle(of a continual octagon)

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

An octagon has 8 sides, so:


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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°

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Interior Angle(of a continual octagon)

Or we can use:


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

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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:

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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

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

Notice that:

the "base" of the triangle is one side of the polygon.the "height" the the triangle is the "Apothem" the the polygon

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:

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(Note: The angles room in radians, no degrees)

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:


sin(π/n) = (Side/2) / RadiusSide = 2 × Radius × sin(π/n)
cos(π/n) = Apothem / RadiusApothem = Radius × cos(π/n)
tan(π/n) = (Side/2) / ApothemSide = 2 × Apothem × tan(π/n)

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:


Area of little Triangle= ½ × Apothem × (Side/2)
TypeName when RegularSides (n) ShapeInterior AngleRadiusSideApothemAreaEquilateral TriangleSquareRegular PentagonRegular HexagonRegular HeptagonRegular Octagon...Regular Pentacontagon
Triangle (or Trigon)3
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60°11.732 (√3)0.51.299 (¾√3)
Quadrilateral(or Tetragon)4
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90°11.414 (√2) 0.707 (1/√2) 2
Pentagon5108°11.1760.8092.378
Hexagon6
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120°110.866 (½√3) 2.598 ((3/2)√3)
Heptagon (or Septagon)7
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128.571°10.8680.9012.736
Octagon8
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135°10.7650.9242.828 (2√2)
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Pentacontagon50172.8°10.126 0.998 3.133
(Note: worths correct to 3 decimal places only)

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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.