📑 Topics on this page (Jump to Topic):
📌 1. Counting Formulas
| Figure | Formula |
|---|---|
| Triangle with lines from apex to base (n lines incl. sides) | n(n−1)/2 |
| Squares in an n × n grid | 1² + 2² + … + n² = n(n+1)(2n+1)/6 |
| Rectangles in an m × n grid (incl. squares) | [m(m+1)/2] × [n(n+1)/2] |
| Square with both diagonals | 8 triangles |
| n points on a line — segments | n(n−1)/2 |
💡 Memory Tip: If no formula fits, count systematically: single parts first, then pairs, then triples… using letter labels.
📌 2. Examples
- 2 × 2 grid: squares = 4 + 1 = 5; rectangles = 3 × 3 = 9.
- 3 × 3 grid: squares = 9 + 4 + 1 = 14; rectangles = 6 × 6 = 36.
- A triangle with 2 inner lines from apex to base (4 lines total) = 4×3/2 = 6 triangles.
⚠️ Exam Traps:
- Rectangle counts include squares — subtract them if the question says 'excluding squares'.
📌 3. Embedded Figures
- The given figure must appear in the complex figure with the same size, angles and orientation — without rotation.
- Find a distinctive corner/angle first, then match the remaining lines.
📌 4. Practice MCQs
📝 Practice MCQs (prepared by SF Prep — not PYQs)
Q1. A square has both diagonals drawn. How many triangles in all? Practice
Correct answer: (B)
4 small + 4 large (two small each) = 8.
4 small + 4 large (two small each) = 8.
Q2. How many squares are in a 3 × 3 grid? Practice
Correct answer: (C)
9 + 4 + 1 = 14.
9 + 4 + 1 = 14.
Q3. Total squares in a 4 × 4 grid? Practice
Correct answer: (D)
16 + 9 + 4 + 1 = 30.
16 + 9 + 4 + 1 = 30.
Q4. How many squares are on an 8 × 8 chessboard? Practice
Correct answer: (A)
8×9×17/6 = 204.
8×9×17/6 = 204.
Q5. Total rectangles (incl. squares) in a 2 × 3 grid? Practice
Correct answer: (B)
(2×3/2) × (3×4/2) = 3 × 6 = 18.
(2×3/2) × (3×4/2) = 3 × 6 = 18.
Q6. Total rectangles (incl. squares) in a 3 × 3 grid? Practice
Correct answer: (C)
6 × 6 = 36.
6 × 6 = 36.
Q7. Rectangles excluding squares in a 3 × 3 grid? Practice
Correct answer: (D)
36 − 14 = 22.
36 − 14 = 22.
Q8. A triangle has 3 inner lines drawn from the apex to the base. Total triangles? Practice
Correct answer: (A)
5 lines in all ⇒ 5×4/2 = 10.
5 lines in all ⇒ 5×4/2 = 10.
Q9. 6 points lie on a line. How many line segments are formed? Practice
Correct answer: (B)
6×5/2 = 15.
6×5/2 = 15.
Q10. Total squares in a 2 × 2 grid? Practice
Correct answer: (C)
4 + 1 = 5.
4 + 1 = 5.
Q11. Total rectangles in a 1 × 4 strip (4 cells in a row)? Practice
Correct answer: (D)
4×5/2 = 10.
4×5/2 = 10.
Q12. Which rule is correct when finding an embedded figure? Practice
Correct answer: (A)
In embedded figures, size and orientation stay the same.
In embedded figures, size and orientation stay the same.
Q13. In a triangle, all three medians are drawn. How many triangles in all? Practice
Correct answer: (B)
6 small + 6 (pairs) + 3 (triples at vertices) + 1 whole = 16.
6 small + 6 (pairs) + 3 (triples at vertices) + 1 whole = 16.
Q14. Total squares in a 5 × 5 grid? Practice
Correct answer: (C)
25 + 16 + 9 + 4 + 1 = 55.
25 + 16 + 9 + 4 + 1 = 55.
Q15. A rectangle has one diagonal drawn. Total triangles? Practice
Correct answer: (D)
The diagonal splits it into 2 triangles.
The diagonal splits it into 2 triangles.
Q16. A triangle has 1 line parallel to the base and 1 line from apex to base. Total triangles? Practice
Correct answer: (A)
3 lines from the apex ⇒ 3 triangles per horizontal level; 2 levels ⇒ 3 × 2 = 6.
3 lines from the apex ⇒ 3 triangles per horizontal level; 2 levels ⇒ 3 × 2 = 6.
Q17. Total rectangles (incl. squares) in a 2 × 4 grid? Practice
Correct answer: (B)
3 × 10 = 30.
3 × 10 = 30.
Q18. A circle has 4 diameters drawn (8 equal parts). How many smallest sectors are there? Practice
Correct answer: (C)
4 diameters = 8 radii ⇒ 8 smallest sectors.
4 diameters = 8 radii ⇒ 8 smallest sectors.
Q19. Total rectangles (incl. squares) in a 2 × 2 grid? Practice
Correct answer: (D)
3 × 3 = 9.
3 × 3 = 9.
Q20. How many diagonals does a pentagon have? Practice
Correct answer: (A)
n(n−3)/2 = 5×2/2 = 5.
n(n−3)/2 = 5×2/2 = 5.