Area Without Integration
The problem
A triangle has its vertices at , and . Count the lattice points strictly inside it and the lattice points on its boundary, then use Pick's theorem to get the area. How many interior lattice points are there, and does the theorem's prediction match a direct count?
- A lattice point has both coordinates integers.
- Boundary points lie on the three edges, vertices included.
Take a minute
Reveal hint 1
Pick's theorem reads , where is the number of interior lattice points and the number on the boundary.
Reveal hint 2
The number of lattice points strictly between two lattice endpoints is .
Solution
Answer interior lattice points; Pick's formula gives , matching the area .
Derivation
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Compute the boundary count.
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Compute the area directly.
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Solve for .
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Verify by listing.
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Understand why the theorem works.