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We have spoken before about polygons, but we have actually not tackled constant polygons by themselves. They are the crème de la crème that polygons, a course unto themselves!

What is a Polygon?

Polygons are straight-sided, level shapes that close in a space. This method every polygon:

Is two-dimensionalUses just line segment to develop its sidesHas one interior and also exterior

The indigenous "polygon" comes from the Greek and method "many angles," since all straightforward polygons have as countless sides as they have actually angles. The most basic polygon, the triangle, has actually three internal angles (and three exterior angles!) therefore it likewise has 3 sides.

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You have the right to have polygons through so plenty of sides, they show up to be circles even when they space not. Once you get above 10 sides, most math students just say "n-gon," whereby n is the variety of sides or internal angles, as in a "18-gon," or a "25-gon," quite than trying to remember the Greek-origin name.

What is a constant Polygon?

To be a regular polygon, the flat, closed, straight-sided form must also have an additional property. Every interior and also exterior edge in the regular polygon is equal to every other interior and also exterior angle, and also every next is same in length to every various other side.


A constant polygon has:

Two dimensionsStraight sidesCongruent (equal-length) sidesAn interior and also exteriorEqual interior anglesEqual exterior angles

Regular Polygon Examples

A few examples of continual polygons are a triangle, quadrilateral, pentagon, hexagon, heptagon, and a decagon.


Regular Vs. Irregular Polygons

If also one of those functions is not present, you perform not have a consistent polygon. Friend may have three that the attributes (two dimensions; directly sides; one interior and exterior) but still not have a continuous polygon. You would have an irregular polygon.


A rectangle that is much longer than the is tall is an example of an rarely often rare polygon. A scalene triangle, house plate ~ above a baseball or soft ball field, and also a kite room all additionally examples of rarely often rare polygons.

Properties of consistent Polygons

Only as soon as all 6 conditions, outlined above, are current will you have actually a continuous polygon. Let"s research a suspicious shape and also see:

Here we have actually △CAT, drawn with black lines. Therefore our black color CAT is suspicious, indeed.


Go v the six properties of regular polygons:

Is the two-dimensional?Is it made with right sides?Are its political parties congruent?Does it close in a space, producing an interior and also exterior?Does it have actually equal inner angles?Does it have equal exterior angles?

Results of ours scrutiny utilizing the 6 identifying nature of consistent polygons: △CAT is a continual polygon, the most basic kind. The is an it is provided triangle, with internal angles the 60° and also exterior angle of 120°.

Names of regular Polygons

Imagine you space unpacking a delivery of polygons. You need to put every the irregular polygons on the discount shelf, and also all the consistent polygons ~ above the full-price shelf. Making use of your knowledge of the identifying properties of consistent polygons, you can easily see that only a few typical polygons are regular:

Regular PolygonNumber the SidesInterior AnglesExterior Angles
Equilateral triangle3 sides3 internal angles that 60°3 exterior angle of 120°
Square4 sides4 inner angles that 90°4 exterior angles of 90°
Regular pentagon5 sides5 inner angles of 108°5 exterior angles of 72°
Regular hexagon6 sides6 internal angles of 120°6 exterior angle of 60°
Regular heptagon7 sides7 interior angles ≅ 128.57°7 exterior angle ≅ 51.43°
Regular octagon8 sides8 inner angles that 135°8 exterior angles of 45°


How plenty of sides does a continuous polygon have?

You can develop regular polygons of any variety of sides and interior angles, yet usually you will certainly not surname them with anything various other than the variety of their political parties or inner angles and also then -gon:

Regular PolygonNumber the SidesInterior AnglesExterior Angles
Regular enneagon or 9-gon9 sides9 internal angles of 140°9 exterior angle of 40°
Regular decagon or 10-gon10 sides10 internal angles that 144°10 exterior angle of 36°
Regular hectogon or 100-gon100 sides100 interior angles of 176.4°100 exterior angle of 3.6°

Any polygon the does not have equal-length sides together with equal interior and exterior angles is an rarely often rare polygon; the goes ~ above the discount shelf!

Parts Of regular Polygons

As basic as they are, consistent polygons still have six parts:

SidesInterior and also exteriorInterior anglesExterior anglesVerticesDiagonals

We will highlight with a an easy regular polygon, the hexagon. This is a familiar shape (think the the cells of a honeybee hive):


Our hexagon has actually six congruent sides. It has actually six same interior angles, each 120°. It also has 6 equal exterior angles. An exterior angle is produced by extending one side and measuring the angle between that extension and an surrounding side.

Regular Polygon Angles

Exterior angle of a continuous Polygon

Exterior angle of every an easy polygon include up come 360°, due to the fact that a trip roughly the polygon completes a rotation, or return to your beginning place. Whereby sides meet, they type vertices, therefore our hexagon likewise has six vertices.

Interior angle of a constant Polygon

Inside the hexagon"s sides, where the inner angles are, is the hexagon"s interior. External its sides is the hexagon"s exterior. This becomes vital when you consider facility polygons, prefer a star-shape (a pentagram, for example).

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Diagonal the a Polygon

Connect any type of non-adjacent vertices in the internal of the hexagon to make diagonals. All simple polygons can be separated into triangles using diagonals. The minimum number of triangles produced in ours hexagon, by illustration three diagonals, is four; every triangle has actually interior angles including to 180°, therefore the amount of the inner angles of a continual hexagon is 4 × 180°, or 720°.

Regular Polygon Formula

The formula because that calculating the sum of interior angles of any kind of regular polygon with n sides is: