CPhill

This will not be a regular octagon

The side of the square is 108/4  =  27

The side of the octagon along the side of the square = 27/3 = 9

And a octagon side adjacent to this one will form an isosceles right triangle with legs of 9

So...the area of one of these triangles is 9^2/2 = 81/2  cm^2

And there will be 4 of these ....one at each vertex of the square

So the total areas of these = 4*81/2 = 162 cm^2   (1)

And the area of the square is 27^2 = 729 cm^2  (2)

So....the area of the octagon is  (2) - (1)  =  567 cm^2 

It looks something like this :

The measure of FAB is just measure of one of the interior angles = 120°

And..in triangle FAB,  AB  and AF  form two equal sides....so the angles opposite these sides are equal.....so....angle AFB = angle ABF

So...

Angle AFB + angle ABF + angle FAB =  180

2 Angle ABF +  120  = 180

2 Angle ABF = 60       divide both sides by 2

Angle ABF =  30°

Rekt.....let's put this in Cartesian form....maybe it will give us a clue as to what the graph might look like

√(x^2+ y^2)  =                4

                             ______________

                              2√(x^2 + y^2) -  x

                             _______________

                                  √(x^2 + y^2)

√ (x^2 + y^2)  =   4 √(x^2 + y^2)

                          _______________

                            2√(x^2 + y^2) - x

1           =               4

                         ______________

                        2√ (x^2 + y^2) - x 

2 √(x^2 + y^2) - x =  4

√(x^2 + y^2) - x/2 = 2

√(x^2 + y^2) =   2 + x/2     square both sides

x^2 + y^2 =   4 + 2x + x^2/4        multiply through by 4   

4x^2 + 4y^2 = 16 + 8x + x^2    simplify

3x^2 - 8x + 4y^2 =16          complete the square on x

3(x^2 - (8/3)x +  16/9   -16/9 ) + 4y^2 = 16

3(x - 4/3)^2 + 4 y^2  =  16 + 48/9

3(x - 4/3)^2  + 4y^2  =  64/3            multiply both sides by 3/64

(9/64) (x - 4/3)^2 + (3/16)y^2  =  1

(x - 4/3)^2             y^2

________   +      _____  =       1

   64/9                  16/3

This is an ellipse centered at (4/3, 0)

a = √[64/ 9] =  8/3

b = √[16/3] =  4/√3

The area is given by

pi * a * b   =

pi * 8/3 * 4/√3    =

pi * 32√3 / 9

k =  32√3 / 9

k^2 =    1024 / 27

Thanks, tertre....!!!

Note......

When θ = 0     r = 3

When θ = pi / 6, r = 0

And this traces out   (1/2) of a "leaf"

So....each pi/6 rads will do the same

So.....6 of these will trace out 3 full "leaves"

So

6 (pi/6)  =   pi  =    M   =   (180”)

r = √ [ x^2 + y^2 ]

cos theta =   x/r =  x / √ [ x^2 + y^2 ]

sin theta = y/r = y / √ [ x^2 + y^2 ]

So we have

√ [ x^2 + y^2 ]   =                 7

                                  _____________

                                    [  3x +  2y ]

                                   ___________

                                   √ [ x^2 + y^2 ]

√ [ x^2 + y^2 ]  =     7 √ [ x^2 + y^2 ]

                              _____________

                                    3x   + 2y

1     =          7

              _______

               3x + 2y

3x + 2y  =   7

In the form Ax + By = C.....the slope is     -A / B

So.....slope   =        -3/2

P(6,2)  = 30

Good Job, Tanmai......!!!!!

"Of" means "multiply"

(2/3) * (15/1)  =  30 / 3  =  10

Let A = drawing a spade from 52 cards

P ( drawing a spade ) = 13/ 52 =  1 / 4    = P(A)

Let B = probability of drawing a spade (after replacement) = 13/52 = 1/4 = P(B)

So

P ( A and B ) = P(A) * P(B)  =  (1/4) * (1/4)  =  1 / 16

6 shows.....choose any 4  =

C(6,4)  =   15   different combos