how is the area of the circle calculated?

Question

A rectangular plywood measures 2.1m by 70 cm . circular discs are to be cut out from the plywood of diameter 14 cm . how many discs can be cut and how much plywood will be left?

Answer 1

 Have to cut Squares of 14cm * 14cm = 196, then inscribe in them circles of diameter = 14cm

 

  Then Numbers of discs will be  14, 700 / 196 = 75 discs

 

   each square of 196 Cm²  , will have  196- 49π = 42.06 as waste.

  

    Total waste = 42.06 * 75 = 3154.5 cm²

 

 

Answer 2

The total area of plywood is 210 cm x 70 cm = 14,700 cm².

The area of each disk is pi*(7 cm)²≈ 154 cm², so the theoretical maximum number of disks is 14,700/154 rounded down to the nearest integer, which is 95. 

 

One can show that the actual maximum number of disks is 77. You get them by cutting in a hexagonal (honeycomb-like) pattern, starting with 5 disks along the 70-cm edge, followed by 4 disks, 5 disks, 4 disks etc. There will be 9 rows where you can cut 5 disks, and 8 rows in between where you can cut 4 disks, for a total of 9x5+8x4=77 disks. You use 77(154 cm²)=11,853 cm² of plywood, so that 14,700-11,853=2847 cm² will be left over.

Answer 3

Hey Ganga -- one method is to line up 5 rows and 15 columns of 14cm discs ...

 

the 75 discs will leave about 20-25% scrap depending on sawdust produced ...

 

thin blade / no sawdust ==> 75 discs leaves about 300 sq.cm scrap.

 

thick blade w/ sawdust may require 5,4,5,4 ... 5 alternating "staggered" columns

 

=> 68 discs, with about 1000 sq.cm more scrap ~ 1300 sq.cm ... Best regards :)