You can already spot a triangle or a square. The real challenge, especially in competitive exams, is finding all the hidden shapes within a complex figure. Simply looking at a diagram and guessing isn't enough. You need a system to make sure you don't miss any.

Let’s start with triangles. One of the most effective methods is vertex-based counting This means you pick one corner (vertex) of the figure and count all the triangles that originate from it. Then, move to the next vertex and do the same, being careful not to count the same triangle twice.

Look at the base of the largest triangle. If it's split into 4 segments, then $n=4$. The total number of triangles would be $4(4+1)/2$, which equals 10. This formula is a huge time-saver.

Counting squares and rectangles inside a grid is another common task. Remember, all squares are rectangles, but not all rectangles are squares. Usually, questions will ask for one or the other specifically.

Let’s tackle squares first. For a square grid of size $n \times n$ (like a chessboard), you can find the total number of squares by summing up the squares of numbers from 1 to $n$. This grid-counting method accounts for squares of all possible sizes, from $1 \times 1$ to the large $n \times n$ square itself.

For a $3 \times 3$ grid, the number of squares is $1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14$.

Counting rectangles in a grid of size $n \times m$ is a bit different. The total number of rectangles is the number of ways to choose two horizontal lines multiplied by the number of ways to choose two vertical lines. This leads to a handy formula.

For our $3 \times 3$ grid, where $n=3$ and $m=3$, the number of rectangles is $(\frac{3(4)}{2}) \times (\frac{3(4)}{2}) = 6 \times 6 = 36$. Notice how there are many more rectangles than squares!

What if the figure isn't a perfect grid or a simple triangle? The key is to break it down. Mentally decompose the complex shape into smaller, familiar shapes. Count the figures in each part separately and then count the figures that are formed by combining these parts.

Always label the regions. Giving each small, indivisible area a name (like A, B, C) can help you track which combinations you've already counted. For example, you can look for all triangles made of one region (A, B, C), then all triangles made of two regions (A+B, B+C), and so on. This makes your count organised and accurate.

Practising these systematic methods is the best way to improve your speed and accuracy. With time, you'll start to see the hidden patterns without even thinking about it.