Counting Figures

Counting Figures involves finding the total number of specific geometric shapes (Triangles, Squares, Rectangles, Lines) hidden within a complex figure.

1. From Vertex: If a triangle is divided by 'n' vertical lines from a vertex, total triangles = Sum of parts = 1 + 2 + ... + (n+1). Correct Formula: If base is divided into n parts, triangles = n(n+1)/2.
2. In a Box: If a square/rectangle is divided by diagonals into 'n' small triangles, total triangles = 2 × n.

1. Squares in NxN Grid: 1² + 2² + ... + N².
2. Squares in MxN Grid: (M×N) + ((M-1)×(N-1)) + ... until one dimension is 1.
3. Rectangles in NxN Grid: (Sum of 1 to N)². Or Cube Sum: 1³ + 2³ + ... + N³.

Count Horizontal (H), Vertical (V), and Slant (S) lines separately. Total = H + V + S.