Shaded Part: Tilted Square
Read the puzzle
A square is divided into a 4 by 4 grid. A tilted gold square sits inside it with one corner on each side: on the top side 3 units from the left, on the right side 1 unit up from the bottom, on the bottom side 1 unit from the left, and on the left side 1 unit down from the top. What fraction of the big square is gold?
What fraction of the square is gold?
Find the area of the four corner triangles that are not gold, then take it away.
5/8
The 4 corner triangles are 3 × 1 ÷ 2 each: 6 of 16 squares. Gold is 10/16 = 5/8.
How to solve it
Do not try to measure the tilted square directly. Find what is not gold and subtract.
The big square holds 4 x 4 = 16 small squares. Outside the gold square there are four dark right triangles, one in each corner. Each one has legs of 3 and 1 units: for example, the top-left corner triangle runs 3 units along the top edge and 1 unit down the left edge. The area of each is 3 x 1 / 2 = 1.5 small squares.
Four of them make 4 x 1.5 = 6 small squares of dark. The gold square is what remains: 16 - 6 = 10 small squares. As a fraction, 10/16 = 5/8.
You can double-check with the side length. Each side of the gold square is the slanted edge of a 3-by-1 triangle, so its length squared is 3 x 3 + 1 x 1 = 10. The area of a square is its side squared, which is 10 again.
The usual wrong answer is 1/2 or 9/16, from counting only the whole gold cells and guessing the cut pieces. Takeaway: for a tilted shape on a grid, subtract the corner triangles.
