How can I check whether I counted every unit square in a shape?

Choose a starting corner and count one row at a time, moving in the same direction across every row. Record each row’s count before moving on. You can lightly mark squares you have checked or number the rows so you can resume if interrupted.

Keep two tallies: squares fully inside the shape, and squares crossed by its boundary. Unit squares are the counting units for area; OpenStax illustrates area by counting equal-sized squares. [S1] Scanning and recording each row is a practical way to apply that idea.

Don’t count a boundary square as a whole square just because most of it is inside. Set it aside for the partial-square step.

Look at the pieces inside the shape and combine pieces that clearly make a whole grid square. For example, two half-squares make one square, and four quarter-squares make one. Keep track of pieces as you pair them so none are used twice. If the pieces do not fit neatly into simple groups, report an estimate rather than claiming an exact count from visual inspection.

Worked example: suppose your row tallies contain 12 full squares. Along the edge, you identify pieces that combine into 6 whole squares. The total is 12 + 6 = 18 square units. If those edge pieces only look approximately equivalent to six squares, write “about 18 square units,” not “exactly 18.”

A hand-drawn, irregular edge can make the partial-square contribution uncertain. Be clear about whether your result is exact or an estimate.

Make a fresh pass with a row-by-row record in front of you. For each row, check that every fully inside square has one tally and that each boundary piece has been recorded once. Compare each row with its first tally, then add the verified row counts again. Checking row by row makes it easier to locate a skipped or duplicated count than checking only the grand total.

For example, if your first pass lists 4, 5, and 3 full squares in three rows, the second pass should confirm those same three counts before you add them. If a row differs, recount that row and inspect its boundary pieces. Don’t rely on recounting the whole shape from memory; you might repeat the same mistake.

Use this routine:

1. Choose a starting corner and scan direction.
2. Record full squares row by row.
3. Add the full-square row counts for a subtotal.
4. Group boundary pieces into whole squares, without reusing any piece; estimate if they do not combine clearly.
5. Add the boundary contribution to the subtotal.
6. Make a fresh pass and verify each row before adding the totals again.

For your notes, write the row counts separately—for example, “Row 1: __; Row 2: __; Row 3: __”—then record the full-square subtotal, boundary contribution, and total. Label the answer “exact” only when the drawing and pieces support an exact count; otherwise, use “about.”