Polygon

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

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Regular

A "Regular Polygon" has:


Otherwise the is irregular.

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Regular PentagonIrregular Pentagon

Here us look in ~ Regular Polygons only.

Properties

So what can we know around regular polygons? an initial of all, we can work out angles.

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Exterior Angle

The Exterior edge is the edge between any type of side of a shape, and also a line expanded from the following side.


AlltheExterior angle of a polygon add up come 360°, so:

Each exterior angle need to 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 angle of a consistent octagon?

An octagon has 8 sides, so:


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Interior Angles

The inner Angle and Exterior Angle room measured from the very same line, therefore they add up to 180°.

Interior angle = 180° − Exterior Angle

We know the Exterior edge = 360°/n, so:

Interior edge = 180° − 360°/n


Example: What is the internal angle that a continuous octagon?

A consistent octagon has actually 8 sides, so:

Exterior edge = 360° / 8 = 45°

internal Angle = 180° − 45° = 135°

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Interior Angle(of a continuous octagon)

Or we can use:


Example: What space the interior and also exterior angle of a regular hexagon?

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A regular hexagon has 6 sides, so:

Exterior edge = 360° / 6 = 60°

interior Angle = 180° − 60° = 120°


And currently for part names:

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

Sounds fairly musical if friend repeat it a couple of times, yet they are just the names of the "outer" and also "inner" circles (and each radius) that deserve to be attracted on a polygon favor this:

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The "outside" circle is dubbed a circumcircle, and also it connects all vertices (corner points) that the polygon.

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

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

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

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

Breaking right into Triangles

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We can learn a lot about regular polygon by breaking them right into triangles favor this:

Notice that:

the "base" that the triangle is one next of the polygon.the "height" of the triangle is the "Apothem" that the polygon

Now, the area the a triangle is fifty percent of the base times height, so:

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

To gain the area of the whole polygon, just include up the areas of every the small triangles ("n" of them):

Area of Polygon = n × next × apothem / 2

And since the perimeter is every the political parties = n × side, we get:

Area the Polygon = perimeter × apothem / 2

A smaller Triangle

By cut the triangle in half we get this:

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(Note: The angles room in radians, not degrees)

The small triangle is right-angled and also so we deserve to use sine, cosine and tangent to uncover how the side, radius, apothem and also n (number the sides) space 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 an ext relationships like those (most of them simply "re-arrangements"), but those will perform for now.

More Area Formulas

We deserve to use that to calculation the area as soon as we only understand 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)
...
Pentacontagon50172.8°10.126 0.998 3.133
(Note: worths correct to 3 decimal places only)

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Graph

And below is a graph that the table above, however with number of sides ("n") indigenous 3 come 30.

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Notice that together "n" it s okay bigger, the Apothem is tending towards 1 (equal to the Radius) and that the Area is tending in the direction of π = 3.14159..., similar to a circle.