Download Interior Angles of Regular Polygons

Interior Angles of Regular Polygons
(Note: it is a Regular Polygon when all sides are equal, all angles are equal.)
If it is a Regular Polygon...
Sum of
Shape
Sides
Shape
Each Angle
Internal Angles
Triangle
3
180°
60°
Quadrilateral
4
360°
90°
Pentagon
5
540°
108°
Hexagon
6
720°
120°
Heptagon (or
Septagon)
7
900°
128.57...°
Octagon
8
1080°
135°
...
...
..
Any Polygon
n
(n-2) × 180°
...
...
(n-2) × 180° / n
That last line can be a bit hard to understand, so let's have one example:
Example: What about a Regular Decagon (10 sides) ?
Sum of Internal Angles = (n-2) × 180°
= (10-2)×180° = 8×180° = 1440°
And it is a Regular Decagon so:
Each internal angle = 1440°/10 = 144°
Dice Probability
Sum of two dice
The first thing is to work out what the range is. You can't have a total less than 2 (both dice being 1) and you can't
have a total more than 12 (both dice being 6). The easiest way to see what the probabilities is to write out the
possible totals. There are 36 of them in all (6 x 6).
Total on dice
2
3
4
5
6
7
8
9
10
11
12
Pairs of dice
Probability
1+1
1/36 = 3%
1+2, 2+1
2/36 = 6%
1+3, 2+2, 3+1
3/36 = 8%
1+4, 2+3, 3+2, 4+1
4/36 = 11%
1+5, 2+4, 3+3, 4+2, 5+1
5/36 = 14%
1+6, 2+5, 3+4, 4+3, 5+2, 6+1 6/36 = 17%
2+6, 3+5, 4+4, 5+3, 6+2
5/36 = 14%
3+6, 4+5, 5+4, 6+3
4/36 = 11%
4+6, 5+5, 6+4
3/36 = 8%
5+6, 6+5
2/36 = 6%
6+6
1/36 = 3%
Either dice is a particular number
The probability of one dice being a particular number is 1/6. You would assume that it would be twice as likely
that either of two dice being a particular number, or 1/3, but this would be wrongWhile there are six conditions
where one dice is a number, and six conditions where the other dice is that number, there is one condition in both.
So the probability is (12-1)/36 or 11/36.
Neither dice is a particular number
The probability of one dice not being a particular number is 5/6. The probability of two dice not being a that
number is 5/6 x 5/6 = 25/36. You can do this as they don't overlap as all (known as an independent event).
Both dice are the same particular number
The probability of one dice being a particular number is 1/6. The probability of two dice being the same particular
number is 1/6 x 1/6 = 1/36. This is not the same as saying that both dice are the same number. There are six
different possible numbers, so that would be 6/36 or 1/6.
At least one dice isn't a particular number
This is the opposite of both dice being the same particular number, so the probabilities will add up to one. So the
probability of at least one dice isn't a particular number = 1 - (1/36) = 35/36.
The dice have two particular numbers
Like the first condition, there are two ways to work this out. The first is doing it directly. Each dice has 6 throws
for each number, but there are four overlaps. So the probability is (4 x 6 - 4)/36 = 20/36.
An easier way is to realise that this condition is the opposite of the next one, dice are not either number. That
probability is 16/36 so this one is 1 - 16/36 = 20/36 or 5/9.
The dice are two particular numbers
There are only two ways that two dice can have two values (try the exercise above to see why), so the probability is
2/36 or 1/18.