Counting Figures
🔒 Log in to trackmedium importance~0-1 Q in Tier 15 formulas⚡ 6 shortcuts3 subtopics
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 extra lines drawn 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, each crossing every apex line.
Counting squares and rectangles
Rectangles in an m × n grid
R = \binom{m+1}{2}\binom{n+1}{2}
Pick 2 of the m + 1 vertical lines and 2 of the n + 1 horizontal lines.
Squares in an m × n grid
S = \sum_{k=1}^{\min(m,n)} (m-k+1)(n-k+1)
An n × n grid gives 1 squared + 2 squared + ... + n squared.
Counting straight lines
Lines in a grid
L=(m+1)+(n+1)
m columns, n rows; add extra slanting lines separately.