Problema Solution
A long time ago Mr Gibson found an island shaped as a triangle with three straight shores of length 3 km, 4km and 5 km. He declared an 'exclusion zone' around his island and forbade anyone to came within 1 km of his shore. What was the area of his exclusion zone ?
Answer provided by our tutors
Make a drawing of the island and the exclusion zone. You can see that the exclusion zone can be seen as composed of 6 pieces:
- Three are rectangular pieces, each 1 km wide and with lengths 3km, 4km, and 5 km.
the are of this 3 pieces is: 3*1 + 4*1 + 5*1 = 12 km^2
- The other three pieces are sectors of circles of radius 1 km.
the sum of the interior angles of a triangle are 180 degrees. that means the sum of the exterior angles is 3x360 - 180 = 900 degrees.
If you subtract the 90 degree "straight" parts (2 per corner), then you have 900 - [3 x 2 x 90] = 900 - 540 = 360 degrees. That's a full circle. So the curved area is r^2*pi = 1^2 * pi = pi km^2.
Thus the exclusion zone is 12 + pi square kilometers.