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Calculating areas on squared paper

Lessons 49–50 · 2 lessons · M.A. Mirzaahmedov, A.A. Rahimqoriyev, Sh.N. Ismailov, M.A. To‘xtaxodjayeva. Mathematics Grade 6, 2nd edition. O‘qituvchi NMIU, Tashkent, 2017
49

Calculating areas on squared paper

Textbook, pp. 217–219
GoalFind the area of a shape on squared paper by counting unit squares and by Pick's formula.
New words
unit square · birlik kvadratlattice point (grid node) · tugun nuqtaPick's formula · Pik formulasiboundary · chegara
Explanation

Measuring the area of a shape means finding how many unit squares it consists of. On squared paper, for a polygon whose vertices are at lattice points (where grid lines cross) we use Pick's formula: S = M/2 + N − 1. Here M is the number of lattice points on the boundary of the polygon and N is the number of lattice points inside it. For example, in a 5 × 4 rectangle M = 18 and N = 12, so S = 18/2 + 12 − 1 = 20, which agrees with 5 · 4 = 20. Pick's formula also works for shapes with slanted sides.

Worked examples
A right triangle with legs 4 and 3 squares: M = 8, N = 3, S = 8/2 + 3 − 1 = 6. Check: 4 · 3 : 2 = 6.
A 6 × 3 rectangle: M = 18, N = 10, S = 9 + 10 − 1 = 18 = 6 · 3.
Class activity

“Count the nodes”: each group draws a polygon on squared paper, counts boundary and interior lattice points, finds the area with Pick's formula and checks by counting squares.

Practice
1
A 6 × 3 rectangle has M = 18 and N = 10. Find its area by Pick's formula.
2
A polygon has 12 boundary and 7 interior lattice points. Find its area.
3
Find the area of a triangle with legs 4 and 3 squares by the ordinary formula (in squares).
4
Why does Pick's formula need both boundary and interior points?