Counting Figures
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Every formula in this topic, grouped by subtopic. Print it and pin it above your desk.
Counting triangles
Apex lines
T = \frac{(k+1)(k+2)}{2}
k lines from the apex to the base.
Apex lines + horizontal cuts
T = \frac{(k+1)(k+2)}{2}\times(h+1)
h lines parallel to the base crossing every apex line.
Counting squares and rectangles
Rectangles in an m × n grid
R = \binom{m+1}{2}\binom{n+1}{2}
Pick two vertical and two horizontal lines.
Squares in an m × n grid
S = \sum_{k=1}^{\min(m,n)} (m-k+1)(n-k+1)
n × n grid: 1² + 2² + … + n².