 Puzzle: What fraction is shaded?

November 1, 2018

There’s nothing like a great geometry puzzle to get the Internet into a frenzy.  Ed Southall, author of “Geometry Snacks,” shared a photo of a pink triangle inside of a square and challenged people to figure out how much of the square is shaded pink.

Here’s the puzzle: What fraction is shaded?

The Solution:

According to Business Insider’s quant reporter Andy Kiersz, the key to solving the problem is the height of the pink triangle.

The area of a triangle is 1/2 (base x height).

If we assume that the square is a 1 x 1 unit, we can see that the base of the pink triangle is 1, the length of the square.

So wll we need to figure out now is the height.

“The key trick is that the little triangle up top is similar to the pink triangle, which means that the little triangle is just a smaller version of the pink triangle,” Kiersz said.

“A property of similar triangles is that the ratio of the triangles’ heights will be the same as the ratio of their bases. Since the pink triangle’s base is twice the little triangle’s base, its height is also twice the little triangle’s height. But we know that the little triangle’s height plus the pink triangle’s height is 1, so that means the pink triangle’s height is 2/3. Plug that on in and we get our area = 1/2 x base x height = 1/2 x 1 x 2/3 = 1/3.”

Hedy Lim‘s solution is neatly written out, which helps to visualize: 