WORKED EXAMPLE · 4 MIN READ

Why a quarter of the height is not a quarter of the area

Compare a rectangle and triangle, calculate a quarter-area cut, and understand absolute percentage-point error.

PulsePuzzle · Reviewed October 13, 2026

SplitPulse asks for a share of area, not a share of width or height. Rectangles make those quantities look interchangeable. Triangles reveal why they are different. This practice example uses simple shapes and a horizontal line; the same reasoning can help you inspect the daily shapes before committing a cut.

Start with a rectangle

Imagine a rectangle 100 units wide and 80 units high. Its area is 100 × 80 = 8,000 square units. A horizontal line 20 units below its top encloses a strip above the line with area 100 × 20 = 2,000. That is 25% of the whole rectangle.

The shortcut works because the strip has the rectangle's full width everywhere. One quarter of the height gives one quarter of the area. It would be a mistake to assume the same relationship for every outline.

Now use a triangle

Imagine a triangle with its point at the top, a horizontal base 100 units wide, and a height of 80 units. Its total area is 100 × 80 ÷ 2 = 4,000 square units. A horizontal line halfway down creates a smaller similar triangle above it.

That small triangle is 40 units high and 50 units wide at its base. Its area is 50 × 40 ÷ 2 = 1,000 square units, exactly 25% of 4,000. The quarter-area cut is therefore halfway down the height, measured from the top point.

Compare two candidate cuts

Both dimensions shrink together in similar triangles. If the height fraction is t, the small triangle's area fraction is t². You do not need to calculate a square root during every round; the half-height/quarter-area reference is already a useful visual anchor.

Score the smaller slice

The same halfway line leaves 25% above and 75% below. SplitPulse always uses the smaller piece, so that cut scores as 25%. Reversing the direction of your drag does not change the result. If a line leaves 40% above and 60% below, its scored slice is 40%, even though the larger region is below.

For a 25% target, achieving a 6.25% smaller slice produces an absolute error of 18.75 percentage points: 25 − 6.25. Achieving 27% produces an error of 2 points. SplitPulse adds the three round errors; an overshoot cannot cancel an undershoot.

Your turn

A cut encloses the top three quarters of the height of a similar triangle. What percentage of the total area lies above it?

Read the SplitPulse guide for line controls and circular outlines, or try the daily area estimates.