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Angles and Polygons lungemine.com Topical summary | JrMath rundown | MathBits" Teacher sources Terms of Use contact Person: Donna Roberts


The interior angles of a polygon are the angle at every vertex ~ above the within of the polygon. In convex polygons, each internal angle is always less 보다 180º


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We have proven that the sum of the measures of the interior angles of a triangle is 180º. We have the right to easily find that xº = 45º.

You are watching: Sum of exterior angles of a hexagon


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A continuous triangle is one equilateral triangle with all sides and also angles of same measure. Us only should divide 180 by 3 to uncover the dimension of every angle.
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When dealing with 4-sided polygons, a diagonal divides the figure into two triangles reflecting that the amount of the interior angles should be 2(180) = 360º.

The strategy presented in the quadrilateral, of dividing the figure into triangles, will certainly be provided to inspection the interior angles of polygons with more than 4 sides. This strategy might be described as "partitioning", "dissecting", or "decomposing".


If we use diagonals come partition a polygon into a series of triangle (as was done v the quadrilateral), we deserve to calculate the amount of the inner angles the larger-sided polygons. How many triangles, developed using diagonals, comprise the polygon?
In a regular hexagon, 4 triangles can be produced using diagonals that the hexagon indigenous a typical vertex. Since the inner angles of each triangle totals 180º, the hexagon"s internal angles will complete 4(180º), or 720º.
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This same technique can be taken in one irregular hexagon. The diagonals type four triangle whose internal angles complete 180º, giving the hexagon"s interior angles a complete of 4(180º), or 720º.
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Notice that the sum of the interior angles is the same for both the regular and the irregular hexagons. additionally notice that we developed 4 triangle in the hexagon (with 6 sides).

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Yes! there is a pattern! The variety of triangles native 1 vertex will certainly be 2 less than the variety of sides, n, that the polygon, or n - 2 triangles. This pattern leader to a formula! The sum that the interior angles is 180º • (the number of triangles formed) or 180•(n - 2).


The pattern emerged in the instance above, is regular (true) for every polygons (both regular and also irregular polygons) .


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(where n = variety of sides)

Special CASE: If you understand that the polygon is a regular polygon, you can find each inner angle by dividing by the variety of sides. (Remember that the interior angles that a continual polygon room equal in measure.)


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You have the right to NOT usage this formula to find each edge in an rarely often, rarely polygon. Because each edge of an rarely often rare polygon may be of different size, over there is no formula for finding individual edge measures.


When working with angle formulas because that polygons, be certain to read closely to determine what you room being asked to find. Look for "hint" words such as sum, interior, exterior, each, degrees and also sides.
Find the variety of degrees in the sum of the interior angle of a decagon.
A decagon has 10 sides. 180(n - 2) = 180(10 - 2) = 180(8) = 1440º.
How plenty of sides walk a polygon have actually if the sum that the interior angle is 1080º?
Set the formula equal to 1080 and also solve for n. 180(n - 2) = 1080 180n - 360 = 1080 180n = 1440 n = 8 (if n is not a confident integer, friend will understand you go something wrong. A polygon cannot have actually a "portion" the a side.)
Find the number of degrees in each interior angle of a regular pentagon.
This is a continual polygon, therefore we deserve to use the formula. A pentagon has actually 5 sides.
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Each interior angle of a regular polygon steps 140º. Find the variety of sides that the polygon.
This is a continuous polygon, so use the formula. Set the formula same to 140 and solve because that n. This solution needs use the algebra skills.
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The exterior edge of a triangle is formed by one side of the triangle and the expansion of an nearby side the the triangle. We will certainly now prolong this ide to polygons.
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In the exact same manner together an exterior edge of a triangle, the exterior edge of a polygon is formed by any type of side that the polygon and the extension of its adjacent side. More formally stated, the exterior angle of a polygonis an edge that creates a linear pair with one of the internal angles that the polygon.


The exterior edge of a polygon is developed by any side of the polygon and also the expansion of its adjacent side. Exterior angles and interior angles room supplementary and type a linear pair. If the polygon is regular, all of its exterior angles will certainly be the same measure.
There space actually two congruent exterior angles at each vertex, but only ONE will be considered for ours work. displayed below, ∠1 and ∠2 space exterior angles and also are likewise congruent upright angles. Keep in mind that ∠3 is not an exterior angle, but it is congruent to inner ∠CDE as they space vertical angles.

If you add ALL exterior angles (ONE at a vertex), friend will acquire a sum of 360º. The sum of the exterior angles of ALL polygon is a constant 360º.


Special CASE: If you know that the polygon is a regular polygon, friend can uncover each exterior angle by splitting by the number of sides, n.(Remember that the exterior angles of a consistent polygon room equal in measure.)


So WHY execute the exterior angles (taken one in ~ a vertex) always add up to 360º nevertheless of the number of sides that the polygon?

Here is what us know around exterior angles and polygons: 1. A polygon with n sides will have n interior angles and n exterior angles (one at every vertex). 2. by its formation, one exterior edge is supplementary to its nearby interior angle. 3. At each vertex the the polygon, the interior angle and also the exterior angle kind a linear pair. Since there are n vertices, there will certainly be n direct pairs in total about the polygon. Each straight pair adds come 180º because that a total of n • 180º or 180n degrees approximately the polygon. 4. us have already shown the the formula for the sum of the interior angles that a polygon v n sides is 180(n - 2). 5. native the amount of ALL linear pairs (180n), subtract the amount of the inner angles (the formula). You will be left with the sum of the exterior angles.

180n - 180(n - 2) = 180n - 180n + 360 = 0 + 360 = 360. The amount of the exterior angles is continuous (360º) for every polygons!!!

Find the sum that the exterior angle of a hexagon. That a pentagon. That a dodecagon. The a quadrilateral. The a 17-gon.
All of these answers room the same. The amount of the exterior angles is 360º.
How countless sides does a polygon have actually if the sum of the exterior angles is 360º?
This is a trick question. The amount of the exterior angles of all polygons is 360º. There is insufficient information to determine the number of sides.
Find the measure up of each exterior edge of a regular octagon.

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This is a consistent polygon, so we can use the formula. An octagon has actually 8 sides. 360/n = 360/8 = 45º
Each exterior angle of a regular polygon has 40º. Find the number of sides that the polygon.
This is a regular polygon, so use the formula. Set the formula same to 40 and solve for n.
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Topical summary | JrMath rundown | lungemine.com | MathBits" Teacher sources Terms that Use contact Person: Donna Roberts