Shaded Part: The Triangle
Read the puzzle
A square is divided into a 4 by 4 grid of equal small squares. A gold triangle is drawn inside it: its base is the whole bottom side of the square, and its top point sits at the middle of the top side. The rest of the square is dark. What fraction of the square is gold?
What fraction of the square is gold?
Cut the triangle down the middle and move one half.
1/2
The triangle has the full base and full height of the square, so it is half: 8 of 16 squares.
How to solve it
Use the grid as a ruler. The big square is 4 small squares wide and 4 tall, so it holds 4 x 4 = 16 small squares.
The gold triangle stands on the full bottom side, so its base is 4 units. Its peak touches the top side, so its height is also 4 units. Triangle area is base x height / 2, which gives 4 x 4 / 2 = 8 small squares.
So the gold part is 8 out of 16, and 8/16 = 1/2. You can also see it without numbers: draw a vertical line down from the peak. Each half of the square is a rectangle cut by the triangle's side into two equal right triangles, one gold and one dark. Gold and dark match in each half, so gold is exactly half.
A common wrong guess is 1/3 or 1/4, because the triangle looks thin near the top and the eye judges by the dark corners, which look large. Another slip is counting only the whole gold squares and ignoring the cut pieces.
Takeaway: any triangle whose base is one full side of a square and whose tip touches the opposite side covers exactly half the square, wherever the tip sits.
