Shaded Part: The Pinwheel
Read the puzzle
A square is divided into a 3 by 3 grid. Four gold triangles form a pinwheel: each has one full side of the square as its base, and its third corner is a corner of the middle grid square, 1 unit in from that side. The dark area is the middle square plus four thin points. What fraction of the square is gold?
What fraction of the square is gold?
Find the dark area instead: one middle square plus four thin slivers.
2/3
The dark part is 1 middle square + 4 halves = 3 of 9 squares, so gold is 6/9 = 2/3.
How to solve it
Measure each gold triangle with the grid. The big square is 3 units on a side, so it holds 3 x 3 = 9 small squares.
Each gold triangle lies along one whole side of the square, so its base is 3 units. Its third corner is a corner of the middle square, which sits 1 unit in from that side, so its height is 1 unit. The area of one triangle is 3 x 1 / 2 = 1.5 small squares.
There are four gold triangles: 4 x 1.5 = 6 small squares. So gold is 6/9 = 2/3.
Check from the dark side. The middle square is 1 small square. Each thin dark point is a triangle with base 1 (a side of the middle square) and height 1, so the four add 4 x 1/2 = 2. Dark is 1 + 2 = 3 small squares, and 9 - 3 = 6 gold, which agrees.
The usual slip is to treat the dark star as large and answer 1/2, or to count only the eight border cells and say 8/9. The slanted edges cut through the cells, so whole-cell counting fails here.
Takeaway: a triangle with height 1 along a side of length 3 always has area 1.5.
