Do equal angles necessarily mean a polygon is regular?Conditions on polygon to ensure equal interior angles or opposite sidesA formula for the exterior angles of an irregular polygon, given interior angles?Regular polygon Interior anglesIf all the 4 sides of a quadrilateral are equal but only 3 of its angles are equal. Is it a square?If I have an $n$-gon, for odd $ngeq5$ with all angles equal to the angles of the regular $n$-gon, is my $n$-gon necessarily regular?Interior angles of irregular polygonn-sided regular polygonIs it possible for a shape to have equal angles but not equal sides/ vice versa?Shape of the limiting polygonIf a quadrilateral has a pair of equal opposite sides, and a pair of equal opposite angles, then is it necessarily a parallelogram?
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Do equal angles necessarily mean a polygon is regular?
Conditions on polygon to ensure equal interior angles or opposite sidesA formula for the exterior angles of an irregular polygon, given interior angles?Regular polygon Interior anglesIf all the 4 sides of a quadrilateral are equal but only 3 of its angles are equal. Is it a square?If I have an $n$-gon, for odd $ngeq5$ with all angles equal to the angles of the regular $n$-gon, is my $n$-gon necessarily regular?Interior angles of irregular polygonn-sided regular polygonIs it possible for a shape to have equal angles but not equal sides/ vice versa?Shape of the limiting polygonIf a quadrilateral has a pair of equal opposite sides, and a pair of equal opposite angles, then is it necessarily a parallelogram?
.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;
$begingroup$
In a polygon, if all the sides are equal, it doesn’t necessarily mean that the polygon is regular (eg. a rhombus). Is this also true for angles? Meaning can you draw a polygon whose interior angles are equal, but the shape is still not regular? I couldn’t think of any examples, but I’m sure there is one.
geometry
asked 9 hours ago
JamminermitJamminermit
15610 bronze badges
$endgroup$
add a comment |
$begingroup$
In a polygon, if all the sides are equal, it doesn’t necessarily mean that the polygon is regular (eg. a rhombus). Is this also true for angles? Meaning can you draw a polygon whose interior angles are equal, but the shape is still not regular? I couldn’t think of any examples, but I’m sure there is one.
geometry
asked 9 hours ago
JamminermitJamminermit
15610 bronze badges
$endgroup$
12
$begingroup$
What about a rectangle?
$endgroup$
– Peter Foreman
9 hours ago
$begingroup$
Wow, what an obvious example I missed! Of course, now I must ask are there any others, an infinite number?
$endgroup$
– Jamminermit
9 hours ago
2
$begingroup$
Any shape with an even number of sides can be made regular and then 'stretched' parallel to one of the sides to create an irregular shape with each angle the same.
$endgroup$
– Peter Foreman
8 hours ago
$begingroup$
Draw a regular polygon. Move one side so it still runs parallel to the original (then another side etc). The equilateral triangle is a special case.
$endgroup$
– Mark Bennet
8 hours ago
$begingroup$
This is, in my opinion, another reason why the triangle is special. Of course in hindsight it's not very surprising, but at least it's something that beyond triangles we need more criteria to assure regularity. In this sense it would seem that there are more $n$-gons (with $n>3$) than triangles, for each $n.$
$endgroup$
– Allawonder
7 hours ago
add a comment |
$begingroup$
In a polygon, if all the sides are equal, it doesn’t necessarily mean that the polygon is regular (eg. a rhombus). Is this also true for angles? Meaning can you draw a polygon whose interior angles are equal, but the shape is still not regular? I couldn’t think of any examples, but I’m sure there is one.
geometry
asked 9 hours ago
JamminermitJamminermit
15610 bronze badges
$endgroup$
In a polygon, if all the sides are equal, it doesn’t necessarily mean that the polygon is regular (eg. a rhombus). Is this also true for angles? Meaning can you draw a polygon whose interior angles are equal, but the shape is still not regular? I couldn’t think of any examples, but I’m sure there is one.
geometry
geometry
asked 9 hours ago
JamminermitJamminermit
15610 bronze badges
asked 9 hours ago
JamminermitJamminermit
15610 bronze badges
asked 9 hours ago
JamminermitJamminermit
15610 bronze badges
asked 9 hours ago
JamminermitJamminermit
15610 bronze badges
asked 9 hours ago
JamminermitJamminermit
15610 bronze badges
15610 bronze badges
12
$begingroup$
What about a rectangle?
$endgroup$
– Peter Foreman
9 hours ago
$begingroup$
Wow, what an obvious example I missed! Of course, now I must ask are there any others, an infinite number?
$endgroup$
– Jamminermit
9 hours ago
2
$begingroup$
Any shape with an even number of sides can be made regular and then 'stretched' parallel to one of the sides to create an irregular shape with each angle the same.
$endgroup$
– Peter Foreman
8 hours ago
$begingroup$
Draw a regular polygon. Move one side so it still runs parallel to the original (then another side etc). The equilateral triangle is a special case.
$endgroup$
– Mark Bennet
8 hours ago
$begingroup$
This is, in my opinion, another reason why the triangle is special. Of course in hindsight it's not very surprising, but at least it's something that beyond triangles we need more criteria to assure regularity. In this sense it would seem that there are more $n$-gons (with $n>3$) than triangles, for each $n.$
$endgroup$
– Allawonder
7 hours ago
add a comment |
12
$begingroup$
What about a rectangle?
$endgroup$
– Peter Foreman
9 hours ago
$begingroup$
Wow, what an obvious example I missed! Of course, now I must ask are there any others, an infinite number?
$endgroup$
– Jamminermit
9 hours ago
2
$begingroup$
Any shape with an even number of sides can be made regular and then 'stretched' parallel to one of the sides to create an irregular shape with each angle the same.
$endgroup$
– Peter Foreman
8 hours ago
$begingroup$
Draw a regular polygon. Move one side so it still runs parallel to the original (then another side etc). The equilateral triangle is a special case.
$endgroup$
– Mark Bennet
8 hours ago
$begingroup$
This is, in my opinion, another reason why the triangle is special. Of course in hindsight it's not very surprising, but at least it's something that beyond triangles we need more criteria to assure regularity. In this sense it would seem that there are more $n$-gons (with $n>3$) than triangles, for each $n.$
$endgroup$
– Allawonder
7 hours ago
12
12
$begingroup$
What about a rectangle?
$endgroup$
– Peter Foreman
9 hours ago
$begingroup$
What about a rectangle?
$endgroup$
– Peter Foreman
9 hours ago
$begingroup$
Wow, what an obvious example I missed! Of course, now I must ask are there any others, an infinite number?
$endgroup$
– Jamminermit
9 hours ago
$begingroup$
Wow, what an obvious example I missed! Of course, now I must ask are there any others, an infinite number?
$endgroup$
– Jamminermit
9 hours ago
2
2
$begingroup$
Any shape with an even number of sides can be made regular and then 'stretched' parallel to one of the sides to create an irregular shape with each angle the same.
$endgroup$
– Peter Foreman
8 hours ago
$begingroup$
Any shape with an even number of sides can be made regular and then 'stretched' parallel to one of the sides to create an irregular shape with each angle the same.
$endgroup$
– Peter Foreman
8 hours ago
$begingroup$
Draw a regular polygon. Move one side so it still runs parallel to the original (then another side etc). The equilateral triangle is a special case.
$endgroup$
– Mark Bennet
8 hours ago
$begingroup$
Draw a regular polygon. Move one side so it still runs parallel to the original (then another side etc). The equilateral triangle is a special case.
$endgroup$
– Mark Bennet
8 hours ago
$begingroup$
This is, in my opinion, another reason why the triangle is special. Of course in hindsight it's not very surprising, but at least it's something that beyond triangles we need more criteria to assure regularity. In this sense it would seem that there are more $n$-gons (with $n>3$) than triangles, for each $n.$
$endgroup$
– Allawonder
7 hours ago
$begingroup$
This is, in my opinion, another reason why the triangle is special. Of course in hindsight it's not very surprising, but at least it's something that beyond triangles we need more criteria to assure regularity. In this sense it would seem that there are more $n$-gons (with $n>3$) than triangles, for each $n.$
$endgroup$
– Allawonder
7 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Start with any polygon that has more than three edges. Move one of the edges parallel to itself a little, extend or contract the adjacent edges appropriately and you will have a new polygon with the same edge directions but different relative side lengths. If you start with a regular polygon the angles will remain all the same.
The idea behind this construction is generic. If you start with any sequence of $n > 3$ vectors that span the plane there will be an $n-2$ dimensional space of linear combinations that vanish. Each such linear combination defines a polygon with the same edge directions: form the partial sums in order to find the vertices.
answered 8 hours ago
Ethan BolkerEthan Bolker
50.9k5 gold badges59 silver badges129 bronze badges
$endgroup$
$begingroup$
I think this strategy works with any polygon that has more than three edges, not just more than four.
$endgroup$
– Kurt Schwanda
8 hours ago
$begingroup$
@KurtSchwanda Fixed, thanks. That's what I meant to say.
$endgroup$
– Ethan Bolker
8 hours ago
$begingroup$
@NateEldredge Done. Note convexity not required.
$endgroup$
– Ethan Bolker
8 hours ago
add a comment |
$begingroup$
Here are four pentagons all with interior angles of $108^circ$. Only the largest is regular. The generalization to any regular polygon should be clear.
answered 8 hours ago
steven gregorysteven gregory
19k3 gold badges27 silver badges60 bronze badges
$endgroup$
add a comment |
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2 Answers
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2 Answers
2
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$begingroup$
Start with any polygon that has more than three edges. Move one of the edges parallel to itself a little, extend or contract the adjacent edges appropriately and you will have a new polygon with the same edge directions but different relative side lengths. If you start with a regular polygon the angles will remain all the same.
The idea behind this construction is generic. If you start with any sequence of $n > 3$ vectors that span the plane there will be an $n-2$ dimensional space of linear combinations that vanish. Each such linear combination defines a polygon with the same edge directions: form the partial sums in order to find the vertices.
answered 8 hours ago
Ethan BolkerEthan Bolker
50.9k5 gold badges59 silver badges129 bronze badges
$endgroup$
$begingroup$
I think this strategy works with any polygon that has more than three edges, not just more than four.
$endgroup$
– Kurt Schwanda
8 hours ago
$begingroup$
@KurtSchwanda Fixed, thanks. That's what I meant to say.
$endgroup$
– Ethan Bolker
8 hours ago
$begingroup$
@NateEldredge Done. Note convexity not required.
$endgroup$
– Ethan Bolker
8 hours ago
add a comment |
$begingroup$
Start with any polygon that has more than three edges. Move one of the edges parallel to itself a little, extend or contract the adjacent edges appropriately and you will have a new polygon with the same edge directions but different relative side lengths. If you start with a regular polygon the angles will remain all the same.
The idea behind this construction is generic. If you start with any sequence of $n > 3$ vectors that span the plane there will be an $n-2$ dimensional space of linear combinations that vanish. Each such linear combination defines a polygon with the same edge directions: form the partial sums in order to find the vertices.
answered 8 hours ago
Ethan BolkerEthan Bolker
50.9k5 gold badges59 silver badges129 bronze badges
$endgroup$
$begingroup$
I think this strategy works with any polygon that has more than three edges, not just more than four.
$endgroup$
– Kurt Schwanda
8 hours ago
$begingroup$
@KurtSchwanda Fixed, thanks. That's what I meant to say.
$endgroup$
– Ethan Bolker
8 hours ago
$begingroup$
@NateEldredge Done. Note convexity not required.
$endgroup$
– Ethan Bolker
8 hours ago
add a comment |
$begingroup$
Start with any polygon that has more than three edges. Move one of the edges parallel to itself a little, extend or contract the adjacent edges appropriately and you will have a new polygon with the same edge directions but different relative side lengths. If you start with a regular polygon the angles will remain all the same.
The idea behind this construction is generic. If you start with any sequence of $n > 3$ vectors that span the plane there will be an $n-2$ dimensional space of linear combinations that vanish. Each such linear combination defines a polygon with the same edge directions: form the partial sums in order to find the vertices.
answered 8 hours ago
Ethan BolkerEthan Bolker
50.9k5 gold badges59 silver badges129 bronze badges
$endgroup$
Start with any polygon that has more than three edges. Move one of the edges parallel to itself a little, extend or contract the adjacent edges appropriately and you will have a new polygon with the same edge directions but different relative side lengths. If you start with a regular polygon the angles will remain all the same.
The idea behind this construction is generic. If you start with any sequence of $n > 3$ vectors that span the plane there will be an $n-2$ dimensional space of linear combinations that vanish. Each such linear combination defines a polygon with the same edge directions: form the partial sums in order to find the vertices.
answered 8 hours ago
Ethan BolkerEthan Bolker
50.9k5 gold badges59 silver badges129 bronze badges
edited 7 hours ago
answered 8 hours ago
Ethan BolkerEthan Bolker
50.9k5 gold badges59 silver badges129 bronze badges
answered 8 hours ago
Ethan BolkerEthan Bolker
50.9k5 gold badges59 silver badges129 bronze badges
answered 8 hours ago
Ethan BolkerEthan Bolker
50.9k5 gold badges59 silver badges129 bronze badges
50.9k5 gold badges59 silver badges129 bronze badges
$begingroup$
I think this strategy works with any polygon that has more than three edges, not just more than four.
$endgroup$
– Kurt Schwanda
8 hours ago
$begingroup$
@KurtSchwanda Fixed, thanks. That's what I meant to say.
$endgroup$
– Ethan Bolker
8 hours ago
$begingroup$
@NateEldredge Done. Note convexity not required.
$endgroup$
– Ethan Bolker
8 hours ago
add a comment |
$begingroup$
I think this strategy works with any polygon that has more than three edges, not just more than four.
$endgroup$
– Kurt Schwanda
8 hours ago
$begingroup$
@KurtSchwanda Fixed, thanks. That's what I meant to say.
$endgroup$
– Ethan Bolker
8 hours ago
$begingroup$
@NateEldredge Done. Note convexity not required.
$endgroup$
– Ethan Bolker
8 hours ago
$begingroup$
I think this strategy works with any polygon that has more than three edges, not just more than four.
$endgroup$
– Kurt Schwanda
8 hours ago
$begingroup$
I think this strategy works with any polygon that has more than three edges, not just more than four.
$endgroup$
– Kurt Schwanda
8 hours ago
$begingroup$
@KurtSchwanda Fixed, thanks. That's what I meant to say.
$endgroup$
– Ethan Bolker
8 hours ago
$begingroup$
@KurtSchwanda Fixed, thanks. That's what I meant to say.
$endgroup$
– Ethan Bolker
8 hours ago
$begingroup$
@NateEldredge Done. Note convexity not required.
$endgroup$
– Ethan Bolker
8 hours ago
$begingroup$
@NateEldredge Done. Note convexity not required.
$endgroup$
– Ethan Bolker
8 hours ago
add a comment |
$begingroup$
Here are four pentagons all with interior angles of $108^circ$. Only the largest is regular. The generalization to any regular polygon should be clear.
answered 8 hours ago
steven gregorysteven gregory
19k3 gold badges27 silver badges60 bronze badges
$endgroup$
add a comment |
$begingroup$
Here are four pentagons all with interior angles of $108^circ$. Only the largest is regular. The generalization to any regular polygon should be clear.
answered 8 hours ago
steven gregorysteven gregory
19k3 gold badges27 silver badges60 bronze badges
$endgroup$
add a comment |
$begingroup$
Here are four pentagons all with interior angles of $108^circ$. Only the largest is regular. The generalization to any regular polygon should be clear.
answered 8 hours ago
steven gregorysteven gregory
19k3 gold badges27 silver badges60 bronze badges
$endgroup$
Here are four pentagons all with interior angles of $108^circ$. Only the largest is regular. The generalization to any regular polygon should be clear.
answered 8 hours ago
steven gregorysteven gregory
19k3 gold badges27 silver badges60 bronze badges
answered 8 hours ago
steven gregorysteven gregory
19k3 gold badges27 silver badges60 bronze badges
answered 8 hours ago
steven gregorysteven gregory
19k3 gold badges27 silver badges60 bronze badges
answered 8 hours ago
steven gregorysteven gregory
19k3 gold badges27 silver badges60 bronze badges
19k3 gold badges27 silver badges60 bronze badges
add a comment |
add a comment |
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12
$begingroup$
What about a rectangle?
$endgroup$
– Peter Foreman
9 hours ago
$begingroup$
Wow, what an obvious example I missed! Of course, now I must ask are there any others, an infinite number?
$endgroup$
– Jamminermit
9 hours ago
2
$begingroup$
Any shape with an even number of sides can be made regular and then 'stretched' parallel to one of the sides to create an irregular shape with each angle the same.
$endgroup$
– Peter Foreman
8 hours ago
$begingroup$
Draw a regular polygon. Move one side so it still runs parallel to the original (then another side etc). The equilateral triangle is a special case.
$endgroup$
– Mark Bennet
8 hours ago
$begingroup$
This is, in my opinion, another reason why the triangle is special. Of course in hindsight it's not very surprising, but at least it's something that beyond triangles we need more criteria to assure regularity. In this sense it would seem that there are more $n$-gons (with $n>3$) than triangles, for each $n.$
$endgroup$
– Allawonder
7 hours ago