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Diagonals Connect two nonconsecutive vertices, and are
drawn with a red dashed line.
Let’s draw all the diagonals from 1 vertex.
Sides # of Triangles Total degrees
5 3 540
Find out how many degrees are in these two shapes, and try to make a formula
Sides # of Triangles Total degrees
7 5 900
n n-2 (n-2)180
5 3 540
6 4 720
Remember, angles on the outside are EXTERIOR ANGLES.
What do all the Exterior Angles of a polygon add up to?
360 degrees!!
What do all the exterior angles of a octagon add up to?
What do all the exterior angles of a decagon add up to?
Theorem 11-1 (Sum of interior angles of polygon) The sum of the measures of the angles of a convex polygon with n
sides is (n-2)180
Theorem 11-2 (Exterior angles sum theorem) The sum of the measure of the exterior angles of a convex polygon
is 360.
What is the measure of one interior angle of a regular pentagon?
5
180)25(
1085
540
What is the measure of one interior angle of a regular octagon?
8
180)28(
1358
1080
The general formula for the measure of one interior angle of a REGULAR polygon is
n
n 180)2(
Fill out this regular polygon chart here.
Sides Name Total interior
Each interior
Total exterior
Each exterior
4
8
12
Think about the relationship between interior and exterior angles.
Interior and exterior angles are supplementary.
n
360
180)2( n
n
n 180)2(
360
Sum of interior angles in polygon
Sum of exterior angles in polygon
Measure of ONE interior angle of REGULAR polygon
Measure of ONE exterior angle of REGULAR polygon
How many sides are there if the one interior angle of a regular polygon is 135 degrees?
How many sides are there if the one exterior angle of a regular polygon is 45 degrees?
Interior and exterior angles are supplementary.
How many sides are there if the one interior angle of a regular polygon is 170 degrees?
How many sides are there if the one exterior angle of a regular polygon is 20 degrees?
Central Angle Angle formed from center of polygon to consecutive vertices.
Apothem Distance from center of polygon to side.
Things to notice, all parts can be found using SOHCAHTOA.It is isosceles, so you can break up the triangle in half.
n
360
Radius
The area of these 5 triangles is =
Or we can think of it as
What do you think we can do to find the area of this shape?
So you see it’s
bhbhbhbhbhA2
1
2
1
2
1
2
1
2
1
PhhbA2
1)5(
2
1
apothemtheisaaPA2
1
Let’s find the area of a pentagon with side length 10
105
Which trig function do we use to find the apothem?
Plug in, be careful with the perimeter!
TANGENT!
a
536tan
72o
36o
8819.6a
)50)(8819.6(2
1A 0477.172
Find the perimeter and area of a rectangle with dimensions:
4 by 10
8 by 20
6 by 15
20 by 50
2 by 5
28
56
42
140
14
40
160
90
1000
10
Side Ratio Perimeter Ratio Area Ratio1:5 1:5 1:254:3 4:3 16:93:1 3:1 9:1
Find the area and perimeter of a rectangle with dimensions:
4 by 10
8 by 20
6 by 15
20 by 50
2 by 5
28
56
42
140
14
40
160
60
1000
10
Side Ratio Perimeter Ratio Area Ratio1:5 1:5 1:254:3 4:3 16:93:1 3:1 9:1
Do you notice a relationship between the side ratio, perimeter ratio, and area ratio? Theorem 11-5
If the scale factor of two similar figures is a:b, then:
1) The ratio of perimeters is a:b
2) The ratio of areas is a2:b2
Two basic problems:
I have two pentagons. If the area of the smaller pentagon is 100, and they have a 1:4 side length ratio, then what is the area of the other pentagon?
I have 2 dodecagons. If the area of one is 314 and the other is 942, what is the side length ratio?
22 4:1
16:1
x
100
16
1
1600x
3:1:
3:1
Two basic problems:
A cracker has a perimeter of 10 inches. A similar mini cracker has perimeter 5 inches. If the area of the regular cracker is 20 in2, what is the area of the mini cracker?
I have 2 n-gons. If the area of one is 135 and the other is 16, what is the perimeter ratio?
Circumference is the distance around the
circle. (Like perimeter)
C = πd = 2πr
Area of a circle:
A = πr2
LIKE THE CRUST
PIZZA PART
.ncecircumfereofPart
lengthABofLength
DEGREESINMEASURED
arcofMeasuremAB
angletheofmeasureisx
rx
ABofLength 2360
x
A
BO
Like crust
Find the Perimeter of this figure.
12
20)20(2)12(288
1664
162440
Do not subtract and then square, must do each circle separately!
Circumference is the distance around the
circle. (Like perimeter)
C = πd = 2πr
Area of a circle:
A = πr2
LIKE THE CRUST
PIZZA PART
Fake sun has a radius of .5 centimeters.
Find the circumference and area of fake sun.
Circumference:
2π(.5) = π
Area:
π(.5)2 = .25π
.ncecircumfereofPart
lengthABofLength
DEGREESINMEASURED
arcofMeasuremAB
angletheofmeasureisx
rx
ABofLength 2360
x
angletheofmeasureisx
rx 2
360AOBsectorofArea
A
BO
Like crust Like the slice