what is the measure to the nearest hundredth of each angle of a regular polygon having seven sides
Interior Angle Sum Theorem
What is true about the sum of interior angles of a polygon ?
The sum of the measures of the interior angles of a convex polygon with n sides is $ (north-ii)\cdot180^{\circ} $
| Shape | Formula | Sum Interior Angles |
|---|---|---|
| $$ \red 3 $$ sided polygon (triangle) | $$ (\red 3-2) \cdot180 $$ | $$ 180^{\circ} $$ |
| $$ \red 4 $$ sided polygon (quadrilateral) | $$ (\ruby 4-2) \cdot 180 $$ | $$ 360^{\circ} $$ |
| $$ \crimson 6 $$ sided polygon (hexagon) | $$ (\red vi-two) \cdot 180 $$ | $$ 720^{\circ} $$ |
Problem one
180°
Yous can also apply Interior Angle Theorem:$$ (\ruby three -2) \cdot 180^{\circ} = (i) \cdot 180^{\circ}= 180 ^{\circ} $$
Problem 2
360° since this polygon is really just ii triangles and each triangle has 180°
You can besides use Interior Bending Theorem:$$ (\red 4 -2) \cdot 180^{\circ} = (two) \cdot 180^{\circ}= 360 ^{\circ} $$
Problem iii
Employ Interior Angle Theorem:$$ (\red 5 -two) \cdot 180^{\circ} = (3) \cdot 180^{\circ}= 540 ^{\circ} $$
Trouble four
Use Interior Angle Theorem: $$ (\red 6 -2) \cdot 180^{\circ} = (iv) \cdot 180^{\circ}= 720 ^{\circ} $$
Video Tutorial
on Interior Angles of a Polygon
Definition of a Regular Polygon:
A regular polygon is but a polygon whose sides all have the aforementioned length and angles all have the same measure. Y'all have probably heard of the equilateral triangle, which are the ii most well-known and near oft studied types of regular polygons.
Examples of Regular Polygons
Regular Hexagon 
Regular Pentagon 
More than on regular polygons here .
Mensurate of a Single Interior Angle
| Shape | Formula | Sum interior Angles |
|---|---|---|
| Regular Pentagon | $$ (\cerise 3-2) \cdot180 $$ | $$ 180^{\circ} $$ |
| $$ \cherry four $$ sided polygon (quadrilateral) | $$ (\scarlet 4-2) \cdot 180 $$ | $$ 360^{\circ} $$ |
| $$ \reddish half dozen $$ sided polygon (hexagon) | $$ (\red vi-2) \cdot 180 $$ | $$ 720^{\circ} $$ |
What almost when you just want 1 interior angle?
In order to find the measure of a unmarried interior angle of a regular polygon (a polygon with sides of equal length and angles of equal measure) with n sides, we summate the sum interior anglesor $$ (\crimson north-ii) \cdot 180 $$ and and then divide that sum by the number of sides or $$ \cerise n$$.
The Formula
The measure of any interior angle of a regular polygon with $$ \red northward $$ sides is
$ \text {any angle}^{\circ} = \frac{ (\red n -ii) \cdot 180^{\circ} }{\reddish n} $
Example 1
Let'south look at an example you're probably familiar with-- the skilful sometime triangle $$\triangle$$ . Now, think this new dominion above only applies to regular polygons. And then, the only type of triangle we could be talking near is an equilateral one like the one pictured beneath. 
Yous might already know that the sum of the interior angles of a triangle measures $$ 180^{\circ}$$ and that in the special case of an equilateral triangle, each angle measures exactly $$ threescore^{\circ}$$.
$ \text{Using our new formula} \\ \text {any angle}^{\circ} = \frac{ (\red northward -2) \cdot 180^{\circ} }{\red northward} \\ \text{ For a triangle , (} \ruby-red 3 \text{ sides)} \\ \frac{ (\red 3 -2) \cdot 180^{\circ} }{\red 3} \\ \frac{ (i) \cdot 180^{\circ} }{\red 3} \\ \frac{180^{\circ}} {\carmine 3} = \fbox{60} $
And so, our new formula for finding the measure of an angle in a regular polygon is consistent with the rules for angles of triangles that we take known from by lessons.
Example 2
To observe the mensurate of an interior angle of a regular octagon, which has 8 sides, apply the formula in a higher place as follows: $ \text{Using our new formula} \\ \text {any angle}^{\circ} = \frac{ (\red n -2) \cdot 180^{\circ} }{\cerise north} \\ \frac{(\red8-2) \cdot 180}{ \red viii} = 135^{\circ} $
Finding ane interior angle of a regular Polygon
Trouble 5
Substitute 8 (an octagon has viii sides) into the formula to detect a single interior bending
Problem 6
Substitute 12 (a dodecagon has 12 sides) into the formula to find a single interior angle
Problem 7
Substitute sixteen (a hexadecagon has sixteen sides) into the formula to find a single interior angle
Claiming Problem
What is the measure of 1 interior angle of a pentagon?
This question cannot be answered because the shape is not a regular polygon. Yous can only use the formula to discover a unmarried interior angle if the polygon is regular!
Consider, for instance, the irregular pentagon below.
You lot can tell, just by looking at the picture, that $$ \angle A and \angle B $$ are not congruent.
The moral of this story- While you can use our formula to observe the sum of the interior angles of any polygon (regular or non), you can not employ this page'south formula for a single angle measure--except when the polygon is regular.
How about the measure of an exterior angle?
Outside Angles of a Polygon
Formula for sum of exterior angles:
The sum of the measures of the exterior angles of a polygon, ane at each vertex, is 360°.
Measure of a Single Exterior Angle
Formula to find 1 angle of a regular convex polygon of due north sides = 
$$ \angle1 + \angle2 + \angle3 = 360° $$
$$ \angle1 + \angle2 + \angle3 + \angle4 = 360° $$
$$ \angle1 + \angle2 + \angle3 + \angle4 + \angle5 = 360° $$
Practise Bug
Problem eight
Substitute 5 (a pentagon has 5sides) into the formula to notice a unmarried outside angle
Problem 9
Substitute 10 (a decagon has 10 sides) into the formula to find a single exterior angle
Problem ten
Substitute 12 (a dodecagon has 12 sides) into the formula to notice a single exterior angle
Challenge Trouble
What is the measure out of 1 exterior angle of a pentagon?
This question cannot be answered because the shape is not a regular polygon. Although you know that sum of the outside angles is 360, you can only utilize formula to find a single exterior angle if the polygon is regular!
Consider, for example, the pentagon pictured beneath. Fifty-fifty though we know that all the outside angles add upwardly to 360 °, we tin see, by but looking, that each $$ \angle A \text{ and } and \angle B $$ are non coinciding..
Make up one's mind Number of Sides from Angles
It's possible to figure out how many sides a polygon has based on how many degrees are in its outside or interior angles.
Trouble xi
Apply formula to find a single exterior angle in reverse and solve for 'n'.
Problem 12
Utilize formula to find a single exterior angle in reverse and solve for 'n'.
Trouble 13
Use formula to observe a single exterior bending in reverse and solve for 'n'.
Claiming Trouble
If each outside angle measures 80°, how many sides does this polygon have?
There is no solution to this question.
When you use formula to find a single outside angle to solve for the number of sides , you get a decimal (4.5), which is incommunicable. Think about it: How could a polygon have 4.5 sides? A quadrilateral has iv sides. A pentagon has v sides.
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Source: https://www.mathwarehouse.com/geometry/polygon/
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