P6: Semi and Quarter Circles

For each figure:
(a) find the diameter of the circles (in centimeters) , and 

(b) perimeter of the figure (in π)

Figure (1) 

Figure (2) 

Solution

Figure (1)

Method 1

The trick for this kind of problem is to observe the picture for the half above and half bottom from point A to point B. In this case:

Half above: d + 12 + d + 12 + d

Half bottom = 22 + d + 26 + d + 22

Length of half above = length of half bottom

==> 3d + 24 = 2d + 60 --> d = 36 cm

Perimeter= 5 semicircles + 84 = 5 * ½π * 36 + 84 = (90π + 84) cm

Method 2

To find the diameter, redraw the problem

It becomes a plain model method

3d + 24 = 2d + 60 --> d = 36 cm

Figure (2)

Let r is the half of diameter (d).

Half above: d + 12 + d + 12 + d

Half bottom = 22 + d + 26 + d + 22

Length of half above = length of half bottom

==> 3r + 14 = 2r + 32 --> r = 18 --> d = 36 cm

Perimeter = 5/4 circle + 5r + 46 = (5/4 * 2 * π * 18) + (5 * 18) + 46 = ( 45π + 136) cm

I can find video explanation here.