SSC CGL Reasoning · Figure Counting
Counting Triangles
Count every triangle in a drawn figure, small and large, by labelling the points and listing the triangles in order, or by splitting a big figure into blocks.
Why this matters
Twenty-five questions in the bank, all from the 2024 shifts, with 4 EASY, 13 MODERATE and 8 HARD. The options sit one apart, so a single missed or doubled triangle costs the mark; a fixed listing order is the only defence.
Concept 1 of 2: Labelling points and listing triangles
Definition
1. Label every corner and every point where two lines cross. 2. Count the single small triangles first, then triangles made of two pieces, then three, and so on up to the whole figure. 3. A set of three points is a triangle only if all three sides are drawn lines and the three points are not on one line.
- Lines from one corner: if lines from the apex of a triangle cut its base into k parts, the figure has k(k + 1) ÷ 2 triangles. One line (2 parts) gives 3; two lines (3 parts) give 6.
- Lines across as well: each line drawn across the triangle, parallel to the base, makes another layer of the same count, so the total is that count times the number of layers.
- A square with both diagonals has 8 triangles: 4 small ones and 4 halves.
- Fast route: spot a standard block (a fan of lines, a crossed square) and use its count, then list only the triangles that use lines outside the block.
Triangles from lines through one corner
- kthe number of parts the base is cut into
Worked example
Practice this conceptself-check · 4 quick reps
The same idea in a real exam question:
Example 1 · Figure Counting · Counting Triangles
Counting only the smallest triangles
Counting three points that lie on one line
Concept 2 of 2: Counting a large figure block by block
Definition
1. Split the figure into blocks you can name: the top-left square, the V at the top, the bottom row. 2. Count the triangles inside each block, using labels where the block is crowded. 3. Add the block counts. 4. Look along each line shared by two blocks for triangles that cross it, and add those.
- A diamond drawn inside a square makes 4 corner triangles; any line that cuts the diamond adds more.
- Fast route: when the options differ by one, recount only the block you were least sure of rather than the whole figure.
Worked example
Practice this conceptself-check · 4 quick reps
The same idea in a real exam question:
Example 2 · Figure Counting · Counting Triangles
Adding the blocks and stopping
Counting a shared triangle in both blocks
Summary — formulas & gotchas at a glance
A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.
Formulas (1)
- Labelling points and listing triangles
Triangles from lines through one corner
Watch out for (4)
- Counting only the smallest triangles→ Labelling points and listing triangles
- Counting three points that lie on one line→ Labelling points and listing triangles
- Adding the blocks and stopping→ Counting a large figure block by block
- Counting a shared triangle in both blocks→ Counting a large figure block by block
Test yourself on Figure Counting
15 past SSC CGL questions from this chapter, timed at 9 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.