Math · Stage 6

Lattice area with Pick's theorem

Don't measure the slanted edges — count the DOTS. Inside dots, plus half the edge dots, take away 1, is the exact area. That little rule is Pick's theorem: A = I + B/2 − 1.

✦ Math📘 Stage 6 · S6⏱ ~14 min🧒 Ages 10–12★ Free to start

I can find a lattice polygon's area with Pick's theorem.

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Kavi's Wonky Shape 🟦

Kavi stretched a rubber band around pegs on a dot board 🟦 to make a lopsided shape with slanty edges. “How do I find its area? I can't count whole squares — the edges cut them into messy bits!” 🤔 Then Kavi grinned: “I'll count the dots inside (that's I), add half the dots on the edge (that's B), and take away 1.” That is Pick's little magic: A = I + B/2 − 1 — exact, every time, slanty edges and all!

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What your child will learn

1Two kinds of dot: inside and on the edgePick's theorem only cares about the dots (the grid points). There are exactly two kinds: I = dots strictly inside the shape, and B = dots sitting right on the boundary (edges and corners). Drop each dot in the right box →

2Pick's theorem: A = I + B/2 − 1Count the inside dots (I), add half the boundary dots (B ÷ 2), then take away 1. Tap each example to hear it — the little working is on the right.

3Predict the area from the dotsYou are told I and B. Work out A = I + B/2 − 1 in your head, then tap your answer.

4Counting squares vs Pick's theoremTap each card. Both try to find the same area — one is painful, one is clean.

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A peek inside

A lattice polygon has 4 interior dots and 8 boundary dots. Find its area.

A = I + B/2 − 1 = 4 + 8/2 − 1 = 4 + 4 − 1

Try this at home: Area: I = 5, B = 8 → A = __

Ready to play and learn?

Start the free interactive lesson, or print the handout for screen-free play.

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Part of Math · Stage 6 · S6 on KAVI Academy — a secular, whole-child learning platform for ages 5–12.