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If the interior angle of a regular polygon is an integer, then so is the exterior angle.
For a regular polygon, the exterior angles are all the same and can be calculated as 360° ÷ number of sides; for this to be an integer, the number of sides must be a factor of 360.
To be a polygon, the shape must have at least 3 sides
So the question is the same as how many factors are there of 360 which are greater than or equal to 3.
360 = 2³ × 3² × 5 → it has (3+1)×(2+1)×(1+1) = 4×3×2 = 24 factors
Of these, the factors 1 (= 2⁰ × 3⁰ × 5⁰) and 2 (= 2¹ × 3⁰ × 5⁰) are less than 3, so there are 24 - 2 = 22 factors greater than or equal to 3.
→ there are 22 regular polygons with integer interior angles.
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The factors of 360 are: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
→The regular polygons with integer interior angles are:
- 3 sides - equilateral triangle
- 4 sides - square
- 5 sides - pentagon
- 6 sides - hexagon
- 8 sides - octagon
- 9 sides - nonagon
- 10 sides - decagon
- 12 sides - duodecagon
- 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360 sides (not really met, so you can look up the names if you are really interested).
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What are regular polygons?
Regular polygons are those polygons that are bothequilateral (all sides congruent) and equilateral (all interior angles congruent).
If a Polygon has odd number of angles angles cannot be congruent?
All regular polygons have equal lengths and equal interior angles but irregular polygons have variations in sizes.
Are all polygons are equilateral?
Yes in general polygons that have equal interior angles also have equal sides and they are said to be regular polygons. But a rectangle is an exception.
What is the formulas to finding interior and exterior angles of regular polygons?
Interior Angles: n-2 (n is number of sides) ____ 180 Exterior angles are always 360 degrees.
Why will only three of the regular polygons tessellate the plane?
The tessellating polygons must meet at a point. At that point, the sum of the interior angles of the polygons must 360 degrees - the sum of angles around any point. Therefore, each interior angle must divide 360 evenly. There is no 1 or 2 sided polygon. The interior angle of a regular pentagon is 108 degrees which does not divide 360 degrees. The interior angles of regular polygons with 7 or more sides lie in the range (120, 180) degrees and so cannot divide 360.That leaves regular polygons with 3, 4 or 6 sides.
What are regular polygons?
Regular polygons are those polygons that are bothequilateral (all sides congruent) and equilateral (all interior angles congruent).
What shape has 45 degree interior angles?
There is no such regular polygon with 45 degree interior angles; the smallest interior angles in regular polygons are 60 degrees, which is found in a triangle.
If a Polygon has odd number of angles angles cannot be congruent?
All regular polygons have equal lengths and equal interior angles but irregular polygons have variations in sizes.
Are all polygons are equilateral?
Yes in general polygons that have equal interior angles also have equal sides and they are said to be regular polygons. But a rectangle is an exception.
What is the formulas to finding interior and exterior angles of regular polygons?
Interior Angles: n-2 (n is number of sides) ____ 180 Exterior angles are always 360 degrees.
What is the measure of an interior angle of a regular polygon with 6 sides?
120See related link below for interior angles of various polygons.
Is there a minimum number of acute interior angles that any polygon can have?
Although a triangle must have at least two acute interior angles, a square has four interior right angles and no acute angles. And as regular polygons have increasing numbers of sides, their interior angles get larger.
Why will only three of the regular polygons tessellate the plane?
The tessellating polygons must meet at a point. At that point, the sum of the interior angles of the polygons must 360 degrees - the sum of angles around any point. Therefore, each interior angle must divide 360 evenly. There is no 1 or 2 sided polygon. The interior angle of a regular pentagon is 108 degrees which does not divide 360 degrees. The interior angles of regular polygons with 7 or more sides lie in the range (120, 180) degrees and so cannot divide 360.That leaves regular polygons with 3, 4 or 6 sides.
Is it true that regular polygons are both equilateral and equiangular?
Yes, regular polygons have equal side lengths and equal interior angles as for example an equilateral triangle or maybe a square.
Explain why regular polygons with more than 6 sides will not tessellate?
The tessellating polygons must meet at a point. At that point, the sum of the interior angles of the polygons must 360 degrees - the sum of angles around any point. Therefore, each interior angle must divide 360 evenly. The interior angles of regular polygons with 7 or more sides lie in the range (120, 180) degrees and so cannot divide 360.
Which polygons cannot be used to form a regular tesselation?
Most regular polygons will not tessellate but if their interior angles is a factor of 360 degrees then they will tessellate or if their angles around a point add up to 360 degrees then they also will tessellate.
What are angles segments triangles etc that have exactly the same measures?
They are said to be regular polygons such as equilateral triangles, squares and other polygons that have congruent sides and equal congruent interior angles.
