Going through the center isn’t enough

Does a line through the middle of a shape always cut it in half? Sometimes. The humble triangle shows how aiming for the center can land you a MISS.

The common belief

Lots of players aim for the middle of the shape, figuring that any line through the center splits it into two equal halves. That’s true for some shapes and false for most of them.

When it works: shapes with a center of symmetry

A circle, an ellipse, a rectangle and a parallelogram all have a center of symmetry: every point has a twin directly opposite, on the other side of the center. A line through that center matches every piece on one side with an identical piece on the other, so it always splits the shape in half, at any angle.

When it fails: the triangle

Take a triangle and its centroid, the point where its three medians meet. Draw a line through that point, parallel to one side. It cuts off a smaller triangle whose sides are two-thirds as long as the big one’s.

So its area is (2/3)² = 4/9 of the total, or 44.44%. The other part is 55.56%. That’s more than 5 points off a 50/50: a perfectly centered cut that still scores a MISS.

The worst it can get

That triangle is actually the worst case. A classic geometry result known as Winternitz’s theorem says that for any convex shape (one with no dents), every line through the centroid leaves at least 4/9 of the area on each side. So aiming at the center of a convex shape guarantees you’ll never do worse than 44.44 / 55.56.

For dented shapes or shapes with holes, all bets are off. The centroid can even land outside the shape entirely, inside the curve of a crescent or in the middle of a hole.

Find the center of mass with cardboard

Cut a shape out of cardboard and poke a small hole near the edge. Hang it from a pushpin through that hole, with a weighted string hanging from the same pin, and trace the line of the string. Do it again from a second hole: the two lines cross at the center of mass. Balance the shape on the tip of a pencil right there and it stays put.

Try it with a triangle, then draw any line through that point and weigh the two pieces. They’re rarely equal, just as the math above predicts.

The takeaway

The center is a good place to start, not the answer. Once you’ve picked your angle, look at which side of your line has more bulk, and shift the line toward it until both halves look balanced.